Browse Data presentation

Totals	Morning	Stem	Evening	Totals
(0)		1	8	(1)
(2)	8 6	2	4 7 9	(3)
(6)	9 7 5 5 3 1	3	0 2 6 6 8	(5)
(10)	9 8 6 4 4 4 4 3 2 0	4	1 3 5 5 7 7 9	(7)
(5)	8 6 5 2 1	5	0 2 3 4 6 8	(6)
(3)	7 3 0	6	1 4 6	(3)
(2)	5 2	7	0 3	(2)
(2)	9 3	8	5	(1)
(0)		9	2 6	(2)

Key:

3 \mid 4 \mid 5

means 43 passengers on a morning crossing and 45 passengers on an evening crossing
(a) Write down the modal number of passengers for these morning crossings.

(1 mark)

Some of the quartiles for these two distributions are shown in the table below.

	Morning	Evening
Lower quartile	$a$	36
Median	$b$	48
Upper quartile	58	$c$

(b) Find the value of

a

, the value of

b

and the value of

c

(3 marks)
(c) For these morning crossings find, to one decimal place,

(i) the mean number of passengers,

(ii) the standard deviation of the number of passengers.

(You may use

\sum x = 1489

and

\sum x^2 = 81135

where

x

is the number of passengers on a morning crossing.)

(3 marks)

One measure of skewness is found using

\frac{3(\text{mean} - \text{median})}{\text{standard deviation}}

(d) Evaluate this measure and describe the skewness of the numbers of passengers on these morning crossings.

(2 marks)
(e) Comment on one difference between the distribution of the numbers of passengers on these morning crossings and the distribution of the numbers of passengers on these evening crossings. State the values of any statistics you have used to support your comment.

(1 mark)

●●●●●Level 410 marksStart

The masses, to the nearest kilogram, of the 36 suitcases checked in for a small charter flight are summarised in the stem and leaf diagram below, where

w

is a constant.

Stem	Leaf	Totals
0	8	(1)
1	1 2 3 4 5 5 6 6 7 8 8 9 9	(13)
2	0 0 1 1 2 2 3 3 4 4 5 6 $w$ $w$ $w$ 9	(16)
3	0 1 2 4 5	(5)
4	7	(1)

Key:

2 \mid 3

means 23 kg
(a) Find the range of these masses.

(1 mark)
(b) Find the median of these masses.

(1 mark)

Given that the interquartile range of these masses is 11.5 kg
(c) find the value of

w

(3 marks)

The airline calculates the following statistics from these masses

\sum x = 804 \qquad \sum x^2 = 20144

(d) Show that the standard deviation of these masses is 7.80 kg to 3 significant figures.

(2 marks)

For these data an outlier is defined as a value that is greater than

Q_3 + 1.5 \times (Q_3 - Q_1)

or smaller than

Q_1 - 1.5 \times (Q_3 - Q_1)

(e) Show that there is exactly one outlier in these data.

(2 marks)
(f) Describe the skewness of these masses. Give a reason for your answer.

(2 marks)

●●●●●Level 411 marksStart

The ages,

a

years, of 120 bicycles brought to a repair workshop are summarised in the table below.

Age ( $a$ years)	Frequency
$0 \leqslant a < 2$	18
$2 \leqslant a < 4$	30
$4 \leqslant a < 5$	24
$5 \leqslant a < 8$	27
$8 \leqslant a < 12$	21

A histogram is drawn to represent these data.

The bar representing the class

2 \leqslant a < 4

is 2.4 cm wide and has an area of 6 cm²
(a) Find the area of the bar representing the class

5 \leqslant a < 8

(2 marks)
(b) Calculate the width and the height of the tallest bar.

(3 marks)
(c) Use linear interpolation to estimate

(i) the median,

(ii) the interquartile range

of the ages of these bicycles.

(4 marks)

One measure of skewness is given by

\frac{Q_3 - 2Q_2 + Q_1}{Q_3 - Q_1}

(d) Evaluate this measure and describe the skewness of these data.

(2 marks)

●●●●●Level 411 marksStart

The lengths,

t

minutes, of 150 calls received by a council helpline on one Monday are summarised in the table below.

Length ( $t$ minutes)	Frequency
$0 \leqslant t < 2$	24
$2 \leqslant t < 5$	45
$5 \leqslant t < 10$	50
$10 \leqslant t < 20$	25
$20 \leqslant t < 30$	6

(a) Give a reason to justify the use of a histogram to represent these data.

(1 mark)
(b) Estimate the number of these calls that lasted between 4 minutes and 12 minutes.

(2 marks)
(c) Use linear interpolation to estimate the median length of these calls.

(2 marks)

You may use

\sum ft^2 = 12\,762.75

(d) Find an estimate for

(i) the mean,

(ii) the standard deviation

of the lengths of these calls.

(4 marks)

Two of these calls are chosen at random.
(e) Estimate the probability that both of these calls lasted between 4 minutes and 12 minutes.

(3 marks)

●●●●●Level 412 marksStart

A baker weighs each loaf of bread from one batch of 20 loaves. The mass of each loaf,

m

grams, is coded using

x = m - 800

and the coded data are summarised by

\sum x = -84 \qquad \sum x^2 = 3406

(a) Find the mean of

m

(3 marks)
(b) Find the standard deviation of

m

(3 marks)

The baker weighs one more loaf from the same batch. For all 21 loaves,

\sum x = -70

(c) Find

(i) the mass of this extra loaf,

(ii) the standard deviation of

x

for all 21 loaves.

(4 marks)

●●●●●Level 410 marksStart

A college records the raw mark,

x

, of each of

12

students on a written chemistry paper and the raw mark,

y

, of each of the same students on a practical chemistry assessment.

Some of the summary statistics are given below.

\sum x = 500 \qquad \sum x^2 = 21\,638 \qquad S_{yy} = 340.92 \qquad S_{xy} = 426.33 \qquad n = 12

(a) (i) Calculate the mean of the

12

raw marks on the written paper.

(ii) Show that the standard deviation of the

12

raw marks on the written paper is

8.19

to 3 significant figures.

(3 marks)
(b) Calculate the product moment correlation coefficient between

x

and

y

(3 marks)

Before publishing the results, the college converts every raw mark into a scaled mark. The formula used is

s = 1.25m + 20

where

m

is the raw mark and

s

is the scaled mark.
(c) Use this formula and the values from part (a) to calculate the mean and the standard deviation of the

12

scaled marks on the written paper. Give your answers to 3 significant figures.

(4 marks)

Given that

$u$ is the scaled mark corresponding to $x$
$v$ is the scaled mark corresponding to $y$

(d) state, giving a reason, the value of the product moment correlation coefficient between

u

and

v

(2 marks)

●●●●●Level 412 marksStart

A farmer records the daily food intake,

x

kg, and the daily milk yield,

y

litres, of each cow in a random sample of

15

of her cows. From the same

15

observations she calculates the following two regression equations.

Regression equation of

y

x

\quad y = 1.5x + 4

Regression equation of

x

y

\quad x = 0.4y + 1.6

(a) (i) Find the point of intersection of these lines.

(ii) Hence show that

\sum x = 120

(4 marks)

Given that

\sum xy = 2130

(b) find

S_{xy}

(4 marks)
(c) Find the product moment correlation coefficient between

x

and

y

(4 marks)

●●●●●Level 412 marksStart

The masses, in grams, of a random sample of 240 potatoes of a certain variety were measured. The results are summarised in the histogram.One of the 240 potatoes is chosen at random, and its mass,

X

grams, is noted.
(a) Show that

\mathrm{P}(160 < X < 180) = 0.183

, correct to 3 significant figures.

(2 marks)

Priya suggests that the distribution of

X

can be well modelled by the distribution

\mathrm{N}(190, 900)

.
(b) (i) Give a brief justification for the use of the normal distribution in this context.

(ii) Give a brief justification for the choice of the parameter values 190 and 900.

(3 marks)
(c) Use Priya's model to find

\mathrm{P}(160 < X < 180)

.

(1 mark)

Omar suggests a different model. He uses the midpoints of the classes to calculate estimates,

m

and

s

, for the mean and standard deviation respectively, in grams, of the 240 masses. He then uses the distribution

\mathrm{N}(m, s^2)

as his model.
(d) Use Omar's model to find

\mathrm{P}(160 < X < 180)

.

(4 marks)

The table shows the probabilities of

X

lying in three further ranges, obtained from the histogram and from the two models. Two of the entries,

p

and

q

, are missing.

	$X < 160$	$180 < X < 220$	$X > 220$
Histogram	0.175	0.458	0.183
Priya's model	$p$	0.472	0.159
Omar's model	0.184	$q$	0.197

(e) (i) Find the values of

p

and

q

.

(ii) By considering the different ranges of values of

X

given in the table, together with your answers to parts (a),
(c) and (d), discuss how well the two models fit the original distribution.

(4 marks)

●●●●●Level 514 marksStart

A technology reviewer tests the battery life of phones of two brands, Brand A and Brand B. The battery life, to the nearest hour, is recorded for 31 phones of Brand A and 27 phones of Brand B, all tested in the same way.

The results are summarised in the back-to-back stem and leaf diagram below, where

k

is a constant.

Totals	Brand A	Stem	Brand B	Totals
(2)	9 4	1	7	(1)
(5)	9 8 7 5 2	2	1 4 6 8	(4)
(10)	8 8 7 6 5 4 3 3 1 0	3	0 1 2 2 3 4 5 5 $k$ 8 9	(11)
(8)	8 7 6 4 3 2 1 0	4	1 3 3 5 6 8	(6)
(4)	6 3 2 0	5	1 3 7	(3)
(1)	1	6	2 6	(2)
(1)	6	7		(0)

Key:

2 \mid 4 \mid 1

means 42 hours for a Brand A phone and 41 hours for a Brand B phone

The median battery life of these Brand B phones is 36 hours.
(a) Find the value of

k

(1 mark)
(b) Find the lower quartile and the upper quartile of the battery life of these Brand A phones.

(2 marks)

The reviewer defines an outlier as an observation that is greater than

Q_3 + 1.5 \times (Q_3 - Q_1)

or less than

Q_1 - 1.5 \times (Q_3 - Q_1)

(c) Show that there is only one outlier amongst the Brand A phones.

(2 marks)

The diagram below shows a box plot for the battery life of these Brand B phones.(d) Using the same scale, draw a box plot for the battery life of these Brand A phones, stating the values that you plot.

(4 marks)
(e) Comment on one difference between the distribution of the battery life of these Brand A phones and the distribution of the battery life of these Brand B phones. State the values of any statistics you have used to support your comment.

(1 mark)

The reviewer realises that the results for 4 more Brand B phones were missed out. In ascending order, the battery lives of these 4 phones, to the nearest hour, are 19 hours,

a

hours, 44 hours and

(2a - 2)

hours.

Given that there is no change to the box plot for the Brand B phones shown in the diagram
(f) find the range of possible values of

a

Show your working clearly.

(3 marks)

●●●●●Level 513 marksStart

A bus company records the journey time, to the nearest minute, of the 07:45 bus from a village to a city centre on 37 weekday mornings during school term time.

The results are summarised in the stem and leaf diagram below.

Stem	Leaf	Totals
2	3	(1)
3	2 4 5 6 7 8 9 9	(8)
4	0 1 1 2 3 3 4 4 5 6 7 8 9	(13)
5	0 1 2 3 3 5 6 8 9	(9)
6	0 2 4 7	(4)
7		(0)
8	1	(1)
9	4	(1)

Key:

4 \mid 6

means 46 minutes
(a) Find

(i) the value of the median

(ii) the value of

Q_1

and the value of

Q_3

for the journey times during term time.

(3 marks)

For these data an outlier is defined as a value that is greater than

Q_3 + 1.5 \times (Q_3 - Q_1)

or smaller than

Q_1 - 1.5 \times (Q_3 - Q_1)

(b) Show that there are 2 outliers for these data.

(3 marks)

The diagram below shows a box plot for the journey times of the same bus on 37 weekday mornings during the school holidays.(c) Using the same scale, draw a box plot to represent the journey times during term time, stating the values that you plot.

(4 marks)
(d) Compare the distribution of the journey times during term time with the distribution of the journey times during the school holidays. You should interpret your comparisons in the context of the question.

(3 marks)

●●●●●Level 513 marksStart

The masses,

w

grams, of 200 strawberries picked on a fruit farm are summarised in the table below.

Mass ( $w$ grams)	Frequency
$5 \leqslant w < 10$	16
$10 \leqslant w < 12$	30
$12 \leqslant w < 14$	52
$14 \leqslant w < 16$	44
$16 \leqslant w < 20$	38
$20 \leqslant w < 30$	20

A histogram is drawn to represent these data.

The tallest bar has a height of 6.5 cm.
(a) Calculate the height of the second tallest bar.

(3 marks)
(b) Estimate the number of these strawberries with a mass between 11 grams and 18 grams.

(2 marks)
(c) Use linear interpolation to estimate the median of

w

(2 marks)

The lower quartile of

w

is 12.15 to 4 significant figures.
(d) Use linear interpolation to estimate

(i) the upper quartile of

w

(ii) the interquartile range of

w

(4 marks)
(e) Describe, giving a reason, the skewness of these data.

(2 marks)

Aisha suggests that a normal distribution could be used to model the masses of these strawberries.
(f) Using your answer to part (e), comment on Aisha's suggestion.

(1 mark)

●●●●●Level 514 marksStart

Measures of spread, standard deviation and coding

38 questions

LessonNot started

The number of times,

x

, that each of

50

households used a recycling centre in a month is summarised below.

\begin{array}{|c|c|c|c|c|c|}\hline x & 0 & 1 & 2 & 3 & 4 \\ \hline \text{Frequency} & 5 & 12 & 18 & 9 & 6 \\ \hline\end{array}

Find the mean and the standard deviation of

x

. (4 marks)

●●●●●Level 24 marksStart

A radio station records the length,

x

seconds, of each of the 40 advertisements that it broadcasts one morning.

The lengths are coded using

y = \frac{x - 30}{5}

and the coded data are summarised by

\sum y = 26 \qquad \sum y^2 = 212

(a) Find the mean of

x

(2 marks)
(b) Find the standard deviation of

x

(3 marks)

The station manager converts each length into minutes.
(c) Find the mean and the standard deviation of the lengths in minutes.

(2 marks)

●●●●●Level 27 marksStart

A data set has smallest value

5

, largest value

30

, lower quartile

12

and upper quartile

20

. Find the range and the interquartile range.

●●●●●Level 22 marksStart

Ffion is planning to write a report on the distances run each week by the members of a running club. There is a large number of members in the club.
(a) State one reason why Ffion might wish to obtain information from a sample of members, rather than from all the members.

(1 mark)
(b) Josh suggests that Ffion should use a sample that is stratified by age group.

Give one advantage of this method as compared with random sampling, in this context.

(1 mark)

Ffion decides to take a random sample of 50 members. She asks each member how far, in kilometres, they ran in the previous week. From her results she calculates the following statistics.

Mean	Standard deviation	Median	Lower quartile	Upper quartile
28.4	9.6	26.0	21.5	33.0

(c) Kofi, a member of the club who was not in the sample, tells Ffion that he ran 50 km in the previous week.

Discuss briefly whether this value should be considered an outlier.

(3 marks)

●●●●●Level 35 marksStart

A company employs

600

staff. A researcher wants a sample of

40

staff to complete a wellbeing survey.
(a) Describe how the researcher could select the sample using systematic sampling. (2 marks)
(b) A colleague suggests instead surveying the first

40

staff to arrive at the office one morning. Explain why this method is likely to produce a biased sample. (2 marks)

The annual salaries, £

x

, of the

40

sampled staff are coded using

y = \dfrac{x - 20000}{1000}

.

The coded values have mean

\bar{y} = 8.4

and standard deviation

s_y = 2.5

.
(c) Find the mean and the standard deviation of the annual salaries. (4 marks)

●●●●●Level 38 marksStart

SECTION B: STATISTICS

Figure 1 shows a cumulative frequency diagram for the times,

t

minutes, taken by

80

students to complete a puzzle.Figure 1
(a) Use Figure 1 to estimate the median time. (2 marks)
(b) Use Figure 1 to estimate the interquartile range of the times. (3 marks)
(c) Estimate the number of students who took more than

25

minutes. (2 marks)

●●●●●Level 37 marksStart

The times,

t

minutes, taken by

60

runners to complete a race are summarised below.

$20 \le t < 24$ : frequency $8$
$24 \le t < 26$ : frequency $14$
$26 \le t < 30$ : frequency $20$
$30 \le t < 38$ : frequency $18$

(a) Use linear interpolation to estimate the median time. (2 marks)
(b) Estimate the mean time. (2 marks)
(c) Estimate the standard deviation of the times. You may use

\sum ft^2 = 49110

. (2 marks)
(d) Describe, with a reason, the skewness of the distribution. (1 mark)

●●●●●Level 37 marksStart

A meteorologist is studying daily maximum temperatures, in °C, recorded at two coastal weather stations,

A

and

B

, on each of the

184

days from 1 May to 31 October in a particular year.
(a) The meteorologist first wants a systematic sample of

23

of the

184

days at station

A

. Describe how such a sample could be taken. (2 marks)

The box plot below summarises the full 184 days at station

A

. The highest temperature recorded was

31.3

°C and the lowest was

11.2

°C.(b) An observation is classed as an outlier if it lies more than

1.5

times the interquartile range above the upper quartile or below the lower quartile. Show that the highest temperature is an outlier but the lowest is not. (3 marks)
(c) For station

B

, the corresponding summary is: lower quartile

15.8

°C, median

17.6

°C, upper quartile

19.7

°C, with no outliers. Compare the distributions of daily maximum temperature at the two stations, making two comparisons in context. (3 marks)

●●●●●Level 38 marksStart

SECTION B: STATISTICS

A researcher records the time,

x

seconds, taken by each of

40

students at a school to complete a puzzle. The times are coded using

y = \dfrac{x - 60}{5}

, and the coded values are summarised by

\sum y = 72 \qquad\qquad \sum y^2 = 394

(a) Find the mean and the standard deviation of the

40

times. (3 marks)
(b) The slowest student took

118

seconds. An outlier is defined as a value more than

2

standard deviations from the mean. Determine whether this time is an outlier. (1 mark)
(c) The

40

students were the first

40

to arrive at the school library one morning. Name this method of sampling, and explain why the sample may not be representative of the students at the school. (2 marks)

●●●●●Level 36 marksStart

Figure 2 shows a box plot for the daily rainfall, in mm, recorded at weather station

A

30

days.Figure 2

For weather station

B

over the same

30

days, the daily rainfall had minimum

0

mm, lower quartile

1.2

mm, median

3.5

mm, upper quartile

7.8

mm and maximum

22

mm.

An outlier is defined as a value more than

1.5 \times

the interquartile range above the upper quartile.
(a) Show that the maximum value of

22

mm at station

B

is an outlier. (2 marks)
(b) Compare the rainfall distributions at the two stations. State three differences. (3 marks)
(c) The rainfall values at station

A

are coded using

y = 2x + 1

, where

x

mm is the recorded rainfall.

Write down the median and the interquartile range of the coded values. (3 marks)

●●●●●Level 38 marksStart

The time taken to complete a 5 kilometre run by each of the 30 members of the Northgate running club can be modelled by a random variable

X

minutes.

It is given that

\sum x = 813 \qquad \sum x^2 = 22\,489

(a) Calculate the mean of

X

(1 mark)
(b) Calculate the standard deviation of

X

(2 marks)
(c) For the members of the Eastbrook running club, the time taken to complete a 5 kilometre run has mean

29.4

minutes and standard deviation

6.3

minutes.

Compare the times taken to complete a 5 kilometre run by the members of the Northgate running club and the Eastbrook running club.

(2 marks)

●●●●●Level 35 marksStart

A gardener grows 12 sunflower seedlings in a new type of compost.

After three weeks, the heights,

h

centimetres, of the seedlings are summarised by

\sum h = 186 \qquad \sum (h - \bar{h})^2 = 97.3

13.2

centimetres and standard deviation

4.6

centimetres.

Compare the heights of the sunflower seedlings grown in the two types of compost.

(2 marks)

●●●●●Level 35 marksStart

The distance,

d

km, that an electric scooter travels on a single charge was recorded for each of 120 scooters.

The histogram below represents these data.The table below shows some of the frequencies.

Distance ( $d$ km)	Frequency
$10 \leqslant d < 20$	15
$20 \leqslant d < 25$	20
$25 \leqslant d < 30$
$30 \leqslant d < 40$	35
$40 \leqslant d < 60$

d

(2 marks)
(d) Find an estimate for the mean of

d

(2 marks)

●●●●●Level 38 marksStart

Volunteers from two teams, team

P

and team

Q

, collect litter from a beach.

The mass,

x

kg, of litter collected by each volunteer is summarised in the table below.

	$n$	$\bar{x}$	$\sum x^2$
Team $P$	16	5.5	530
Team $Q$	24	$b$	1071

The two teams are combined into one group of 40 volunteers. The mean mass of litter collected by all 40 volunteers is 5.95 kg.
(a) Show that

b = 6.25

(2 marks)
(b) Find the standard deviation of the masses of litter collected by all 40 volunteers.

(2 marks)

It is later found that the scales used by team

Q

read 0.4 kg too low, so 0.4 kg is added to the mass recorded for each volunteer in team

Q

.
(c) Without further calculations state, giving a reason in each case, the effect this will have on

(i) the variance of the masses for team

Q

(ii) the mean mass for all 40 volunteers

(iii) the standard deviation of the masses for all 40 volunteers.

(3 marks)

●●●●●Level 37 marksStart

Find the variance and standard deviation of the data

2, 4, 4, 4, 5, 5, 7, 9

●●●●●Level 33 marksStart

For a data set of

10

values,

\sum x = 120

and

\sum x^2 = 1640

. Find the standard deviation, to 3 s.f.

●●●●●Level 32 marksStart

The table shows the number of pets owned by

20

people.

\begin{array}{c|cccc} x & 0 & 1 & 2 & 3 \\\\ \hline f & 5 & 8 & 4 & 3 \end{array}

Find the standard deviation, to 3 s.f.

●●●●●Level 33 marksStart

A data set has lower quartile

20

and upper quartile

40

. An outlier is a value more than

1.5\times\text{IQR}

beyond a quartile.
(a) Find the boundaries beyond which a value is an outlier.
(b) State whether

75

is an outlier.

●●●●●Level 33 marksStart

The variable

x

is coded using

y = \dfrac{x - 50}{10}

. The standard deviation of the

y

-values is

1.4

. Find the standard deviation of the

x

-values.

●●●●●Level 32 marksStart

The stem-and-leaf diagram shows the lengths, in minutes, of the 19 telephone calls made from a small office on one day.

Stem	Leaves
0	4
1
2	3 6 8
3	0 1 4 5 5 7 9
4	1 2 2 6 8
5	0 3 8

Key:

2 \mid 3

●●●●●Level 47 marksStart

The stem-and-leaf diagram shows the times, in minutes, taken by 21 people to complete a charity fun run, measured correct to the nearest minute.

Stem	Leaves
2	2 5 7 8 9
3	1 1 3 4 5 6 6 8 9
4	0 1 3 4 7
5	2
6
7
8	1

Key:

3 \mid 4

●●●●●Level 46 marksStart

As part of a study, the masses,

m

grams, of a random sample of 600 eggs from one farm were recorded. The results are summarised in the table.

Mass (g)	$45 \leqslant m < 55$	$55 \leqslant m < 60$	$60 \leqslant m < 65$	$65 \leqslant m < 75$
Frequency	88	214	236	62

(a) Calculate estimates of the mean and standard deviation of these masses.

(2 marks)

The masses,

x

grams, of a random sample of 600 eggs from a second farm were also found. It was proposed that the distribution of the masses of eggs from this farm should be modelled by the random variable

X \sim \mathrm{N}(61, 36)

.
(b) Use the model to find

\mathrm{P}(55 < X < 64)

.

(1 mark)
(c) Use the model to determine

x_1

such that

\mathrm{P}(58 < X < x_1) = 0.5

●●●●●Level 49 marksStart

Figure 2

Figure 2 is the cumulative frequency graph for the times,

t

minutes, taken by

60

people to complete a task.
(a) Use the graph to estimate the median and the interquartile range. (4 marks)
(b) An outlier is defined as a value more than

1.5 \times

the interquartile range above the upper quartile or below the lower quartile. Show that this data set contains no outliers. (2 marks)

●●●●●Level 46 marksStart

The heights, in cm, of a random sample of plants of variety

A

are summarised below.

\begin{array}{|c|c|c|c|c|}\hline \text{Minimum} & \text{Lower quartile} & \text{Median} & \text{Upper quartile} & \text{Maximum} \\ \hline 8 & 14 & 19 & 25 & 45 \\ \hline \end{array}

An outlier is defined as a value that is more than

1.5 \times

the interquartile range above the upper quartile, or more than

1.5 \times

the interquartile range below the lower quartile.
(a) Show that the maximum height of

45

cm is an outlier. (2 marks)

Figure 2 shows a box plot for the heights of a random sample of plants of variety

B

.Figure 2
(b) Compare the distributions of the heights of the two varieties. State three differences. (3 marks)
(c) The heights of variety

A

are coded using

y = \dfrac{x - 10}{5}

, where

x

cm is the original height.

Find the median and the interquartile range of the coded heights. (3 marks)
(d) State, giving a reason, whether the heights of variety

A

are positively or negatively skewed. (1 mark)

●●●●●Level 49 marksStart

Totals	Morning	Stem	Evening	Totals
(0)		1	8	(1)
(2)	8 6	2	4 7 9	(3)
(6)	9 7 5 5 3 1	3	0 2 6 6 8	(5)
(10)	9 8 6 4 4 4 4 3 2 0	4	1 3 5 5 7 7 9	(7)
(5)	8 6 5 2 1	5	0 2 3 4 6 8	(6)
(3)	7 3 0	6	1 4 6	(3)
(2)	5 2	7	0 3	(2)
(2)	9 3	8	5	(1)
(0)		9	2 6	(2)

Key:

3 \mid 4 \mid 5

	Morning	Evening
Lower quartile	$a$	36
Median	$b$	48
Upper quartile	58	$c$

(b) Find the value of

a

, the value of

b

and the value of

c

(3 marks)
(c) For these morning crossings find, to one decimal place,

(i) the mean number of passengers,

(ii) the standard deviation of the number of passengers.

(You may use

\sum x = 1489

and

\sum x^2 = 81135

where

x

is the number of passengers on a morning crossing.)

(3 marks)

One measure of skewness is found using

\frac{3(\text{mean} - \text{median})}{\text{standard deviation}}

●●●●●Level 410 marksStart

The masses, to the nearest kilogram, of the 36 suitcases checked in for a small charter flight are summarised in the stem and leaf diagram below, where

w

is a constant.

Stem	Leaf	Totals
0	8	(1)
1	1 2 3 4 5 5 6 6 7 8 8 9 9	(13)
2	0 0 1 1 2 2 3 3 4 4 5 6 $w$ $w$ $w$ 9	(16)
3	0 1 2 4 5	(5)
4	7	(1)

Key:

2 \mid 3

w

(3 marks)

The airline calculates the following statistics from these masses

\sum x = 804 \qquad \sum x^2 = 20144

(d) Show that the standard deviation of these masses is 7.80 kg to 3 significant figures.

(2 marks)

For these data an outlier is defined as a value that is greater than

Q_3 + 1.5 \times (Q_3 - Q_1)

or smaller than

Q_1 - 1.5 \times (Q_3 - Q_1)

(e) Show that there is exactly one outlier in these data.

(2 marks)
(f) Describe the skewness of these masses. Give a reason for your answer.

(2 marks)

●●●●●Level 411 marksStart

The ages,

a

years, of 120 bicycles brought to a repair workshop are summarised in the table below.

Age ( $a$ years)	Frequency
$0 \leqslant a < 2$	18
$2 \leqslant a < 4$	30
$4 \leqslant a < 5$	24
$5 \leqslant a < 8$	27
$8 \leqslant a < 12$	21

A histogram is drawn to represent these data.

The bar representing the class

2 \leqslant a < 4

is 2.4 cm wide and has an area of 6 cm²
(a) Find the area of the bar representing the class

5 \leqslant a < 8

\frac{Q_3 - 2Q_2 + Q_1}{Q_3 - Q_1}

(d) Evaluate this measure and describe the skewness of these data.

(2 marks)

●●●●●Level 411 marksStart

The lengths,

t

minutes, of 150 calls received by a council helpline on one Monday are summarised in the table below.

Length ( $t$ minutes)	Frequency
$0 \leqslant t < 2$	24
$2 \leqslant t < 5$	45
$5 \leqslant t < 10$	50
$10 \leqslant t < 20$	25
$20 \leqslant t < 30$	6

\sum ft^2 = 12\,762.75

●●●●●Level 412 marksStart

A baker weighs each loaf of bread from one batch of 20 loaves. The mass of each loaf,

m

grams, is coded using

x = m - 800

and the coded data are summarised by

\sum x = -84 \qquad \sum x^2 = 3406

(a) Find the mean of

m

(3 marks)
(b) Find the standard deviation of

m

(3 marks)

The baker weighs one more loaf from the same batch. For all 21 loaves,

\sum x = -70

(c) Find

(i) the mass of this extra loaf,

(ii) the standard deviation of

x

for all 21 loaves.

(4 marks)

●●●●●Level 410 marksStart

A company asked

11

of its employees for the distance,

d

km, from home to the office and the time,

t

minutes, of their journey to work on one morning. The results are shown in the table below.

Employee	A	B	C	D	E	F	G	H	I	J	K
Distance ( $d$ km)	3	5	6	8	9	11	12	14	17	19	22
Time ( $t$ minutes)	12	15	21	20	95	26	28	31	33	40	43

On that morning, employee E was delayed by a cancelled train.

An outlier is defined as a value that is greater than

Q_3 + 1.5 \times (Q_3 - Q_1)

or smaller than

Q_1 - 1.5 \times (Q_3 - Q_1)

(a) Show that

95

is an outlier for the journey times.

(3 marks)

Leaving out employee E, the company calculated the following summary statistics for the other

10

employees.

\sum d = 117 \qquad \sum t = 269 \qquad S_{dd} = 360.1 \qquad S_{dt} = 572.7

(b) Use these summary statistics to show that the equation of the least squares regression line of

t

d

for these

10

employees is

t = 8.29 + 1.59d

where the values of the intercept and gradient are given to 3 significant figures. You must show your working.

(3 marks)
(c) Give an interpretation of the gradient of the regression line.

(1 mark)

Two new employees live

16

km and

30

km from the office.
(d) Using the equation given in part (b), estimate the journey time for

(i) the employee who lives

16

km from the office,

(ii) the employee who lives

30

km from the office.

(3 marks)
(e) State, giving a reason, which of the two estimates found in part (d) would be the more reliable estimate.

(2 marks)

●●●●●Level 412 marksStart

A college records the raw mark,

x

, of each of

12

students on a written chemistry paper and the raw mark,

y

, of each of the same students on a practical chemistry assessment.

Some of the summary statistics are given below.

\sum x = 500 \qquad \sum x^2 = 21\,638 \qquad S_{yy} = 340.92 \qquad S_{xy} = 426.33 \qquad n = 12

(a) (i) Calculate the mean of the

12

raw marks on the written paper.

(ii) Show that the standard deviation of the

12

raw marks on the written paper is

8.19

to 3 significant figures.

(3 marks)
(b) Calculate the product moment correlation coefficient between

x

and

y

(3 marks)

Before publishing the results, the college converts every raw mark into a scaled mark. The formula used is

s = 1.25m + 20

where

m

is the raw mark and

s

is the scaled mark.
(c) Use this formula and the values from part (a) to calculate the mean and the standard deviation of the

12

scaled marks on the written paper. Give your answers to 3 significant figures.

(4 marks)

Given that

$u$ is the scaled mark corresponding to $x$
$v$ is the scaled mark corresponding to $y$

(d) state, giving a reason, the value of the product moment correlation coefficient between

u

and

v

(2 marks)

●●●●●Level 412 marksStart

k

is a constant.

Totals	Brand A	Stem	Brand B	Totals
(2)	9 4	1	7	(1)
(5)	9 8 7 5 2	2	1 4 6 8	(4)
(10)	8 8 7 6 5 4 3 3 1 0	3	0 1 2 2 3 4 5 5 $k$ 8 9	(11)
(8)	8 7 6 4 3 2 1 0	4	1 3 3 5 6 8	(6)
(4)	6 3 2 0	5	1 3 7	(3)
(1)	1	6	2 6	(2)
(1)	6	7		(0)

Key:

2 \mid 4 \mid 1

means 42 hours for a Brand A phone and 41 hours for a Brand B phone

The median battery life of these Brand B phones is 36 hours.
(a) Find the value of

k

(1 mark)
(b) Find the lower quartile and the upper quartile of the battery life of these Brand A phones.

(2 marks)

The reviewer defines an outlier as an observation that is greater than

Q_3 + 1.5 \times (Q_3 - Q_1)

or less than

Q_1 - 1.5 \times (Q_3 - Q_1)

a

hours, 44 hours and

(2a - 2)

hours.

Given that there is no change to the box plot for the Brand B phones shown in the diagram
(f) find the range of possible values of

a

Show your working clearly.

(3 marks)

●●●●●Level 513 marksStart

Stem	Leaf	Totals
2	3	(1)
3	2 4 5 6 7 8 9 9	(8)
4	0 1 1 2 3 3 4 4 5 6 7 8 9	(13)
5	0 1 2 3 3 5 6 8 9	(9)
6	0 2 4 7	(4)
7		(0)
8	1	(1)
9	4	(1)

Key:

4 \mid 6

means 46 minutes
(a) Find

(i) the value of the median

(ii) the value of

Q_1

and the value of

Q_3

for the journey times during term time.

(3 marks)

For these data an outlier is defined as a value that is greater than

Q_3 + 1.5 \times (Q_3 - Q_1)

or smaller than

Q_1 - 1.5 \times (Q_3 - Q_1)

●●●●●Level 513 marksStart

The masses,

w

grams, of 200 strawberries picked on a fruit farm are summarised in the table below.

Mass ( $w$ grams)	Frequency
$5 \leqslant w < 10$	16
$10 \leqslant w < 12$	30
$12 \leqslant w < 14$	52
$14 \leqslant w < 16$	44
$16 \leqslant w < 20$	38
$20 \leqslant w < 30$	20

w

(2 marks)

The lower quartile of

w

is 12.15 to 4 significant figures.
(d) Use linear interpolation to estimate

(i) the upper quartile of

w

(ii) the interquartile range of

w

●●●●●Level 514 marksStart

An auction house sells used cars. Before each sale, the auction house gives each car a guide price, £

x

thousand. For a random sample of

10

cars, the guide price and the sale price, £

y

thousand, were recorded. The guide prices of these cars were between £

4

thousand and £

31

thousand.

The data are summarised as follows

\sum x = 150 \qquad \sum y = 134.7 \qquad \sum y^2 = 2242.73 \qquad \sum xy = 2584.4 \qquad S_{xx} = 746

(a) Calculate the exact value of

S_{xy}

and the exact value of

S_{yy}

(3 marks)
(b) Calculate the value of the product moment correlation coefficient between

x

and

y

(2 marks)
(c) Give an interpretation of your product moment correlation coefficient.

(1 mark)
(d) Show that the equation of the regression line of

y

x

can be written as

y = 2.13 + 0.756x

where the values of the intercept and gradient are given to 3 significant figures.

(3 marks)
(e) Give an interpretation, in context, of the gradient of the regression line.

(1 mark)

Using the equation of the regression line given in part (d)
(f) (i) estimate the sale price of a car with a guide price of £

10\,000

(ii) explain why an estimate of the sale price of a car with a guide price of £

60\,000

is not reliable.

(2 marks)

The auction house says that a car has sold "below guide" if its sale price is less than

90\%

of its guide price.
(g) Using the equation of the regression line given in part (d), find the range of values of

x

for which a car is expected to sell below guide.

(2 marks)

●●●●●Level 514 marksStart

Statistical models allow a complicated real-world situation to be described in a simpler way.
(a) Give one other reason why statistical models are used.

(1 mark)

Kofi has solar panels on the roof of his house. He wants to model the relationship between the number of hours of sunshine in a day,

s

, and the amount of electricity generated by the panels that day,

g

kWh.

Kofi takes a random sample of

12

days and codes the amount of electricity generated so that

w = 20g

These data are summarised as follows

S_{ss} = 140.8425 \qquad S_{sw} = 1881.35 \qquad \sum s = 79.5 \qquad \sum w = 1570 \qquad \sum w^2 = 232\,756

(b) Show that

S_{ww} = 27\,347.67

to 2 decimal places.

(1 mark)
(c) Find the value of

S_{sg}

and the value of

S_{gg}

(3 marks)
(d) Find the product moment correlation coefficient between

g

and

s

(2 marks)
(e) Give an interpretation, in context, of your product moment correlation coefficient.

(1 mark)
(f) Show that the equation of the regression line of

w

s

w = 42.3 + 13.4s

where the values of the intercept and the gradient are given to 3 significant figures.

(3 marks)
(g) Write down an equation of the regression line of

g

s

(1 mark)
(h) Using your equation in part (g)

(i) estimate the amount of electricity generated on a day with

7

hours of sunshine,

(ii) interpret the effect that an increase of

1

hour of sunshine is expected to have on the amount of electricity generated.

(2 marks)

●●●●●Level 514 marksStart

A garden centre is investigating the growth of a variety of sunflower. For each of

12

seedlings, the number of days after planting,

x

, and the height of the seedling,

y

cm, are recorded, where

5 \leqslant x \leqslant 40

The regression line of

y

x

is found to be

y = -6.7 + 1.1x

(a) Give an interpretation of the gradient of the regression line.

(1 mark)
(b) Use the regression line to estimate the height of a seedling

55

days after planting.

(2 marks)
(c) Comment on the reliability of your estimate, giving a reason for your answer.

(2 marks)

Using the regression line of

y

x

and the following summary statistics

\sum y = 212.2 \qquad \sum y^2 = 5606.62 \qquad \sum x^2 = 7378 \qquad n = 12

(d) show that the product moment correlation coefficient for these data is

0.983

to 3 decimal places.

(7 marks)

The garden centre manager plots the data on the scatter diagram below.With reference to both the scatter diagram and the correlation coefficient,
(e) discuss the suitability of a linear regression model to describe the relationship between

x

and

y

.

(2 marks)

●●●●●Level 514 marksStart

In a particular year, the wingspan of an adult barn owl ringed at a wildlife reserve, Reserve A, has a mean of

88.0

centimetres and a standard deviation of

2.4

centimetres.

The wingspans of

95\%

of these owls are between

83.3

centimetres and

92.7

centimetres.
(a) Comment on whether a normal distribution may be suitable to model the wingspan of an adult barn owl ringed at Reserve A in this particular year.

(3 marks)
(b) You may assume that the wingspan of an adult barn owl ringed at Reserve A may be modelled by a normal distribution with mean

88.0

centimetres and standard deviation

2.4

centimetres.

(i) Find the probability that the wingspan of a randomly selected adult barn owl ringed at Reserve A is

90

centimetres.

(ii) Find the probability that the wingspan of a randomly selected adult barn owl ringed at Reserve A is between

85

centimetres and

91

centimetres.

(iii) Two adult barn owls ringed at Reserve A are chosen at random.

Calculate the probability that both of their wingspans are between

85

centimetres and

91

centimetres.

(3 marks)
(c) The summarised data for the wingspans,

w

centimetres, of a random sample of

25

adult barn owls ringed at a second reserve, Reserve B, is given below.

\sum w = 2127.5 \qquad \sum (w - \bar{w})^2 = 230.64

Use this data to calculate estimates of the mean and standard deviation of the wingspans of adult barn owls ringed at Reserve B.

(3 marks)
(d) Using your answers from part (c), compare the wingspans of adult barn owls ringed at Reserve A and adult barn owls ringed at Reserve B.

(2 marks)

●●●●●Level 511 marksStart

Cumulative frequency, box plots and outliers

18 questions

LessonNot started

SECTION B: STATISTICS

Figure 1 shows a cumulative frequency diagram for the times,

t

minutes, taken by

80

25

minutes. (2 marks)

●●●●●Level 37 marksStart

A meteorologist is studying daily maximum temperatures, in °C, recorded at two coastal weather stations,

A

and

B

, on each of the

184

days from 1 May to 31 October in a particular year.
(a) The meteorologist first wants a systematic sample of

23

of the

184

days at station

A

. Describe how such a sample could be taken. (2 marks)

The box plot below summarises the full 184 days at station

A

. The highest temperature recorded was

31.3

°C and the lowest was

11.2

°C.(b) An observation is classed as an outlier if it lies more than

1.5

times the interquartile range above the upper quartile or below the lower quartile. Show that the highest temperature is an outlier but the lowest is not. (3 marks)
(c) For station

B

, the corresponding summary is: lower quartile

15.8

°C, median

17.6

°C, upper quartile

19.7

°C, with no outliers. Compare the distributions of daily maximum temperature at the two stations, making two comparisons in context. (3 marks)

●●●●●Level 38 marksStart

Figure 2 shows a box plot for the daily rainfall, in mm, recorded at weather station

A

30

days.Figure 2

For weather station

B

over the same

30

days, the daily rainfall had minimum

0

mm, lower quartile

1.2

mm, median

3.5

mm, upper quartile

7.8

mm and maximum

22

mm.

An outlier is defined as a value more than

1.5 \times

the interquartile range above the upper quartile.
(a) Show that the maximum value of

22

mm at station

B

is an outlier. (2 marks)
(b) Compare the rainfall distributions at the two stations. State three differences. (3 marks)
(c) The rainfall values at station

A

are coded using

y = 2x + 1

, where

x

mm is the recorded rainfall.

Write down the median and the interquartile range of the coded values. (3 marks)

●●●●●Level 38 marksStart

A cumulative frequency graph for

80

values is used to estimate the median. The median is the value at a cumulative frequency of how many, and which class does it lie in if the cumulative frequency reaches

35

x = 20

and

52

x = 30

●●●●●Level 32 marksStart

A cumulative frequency distribution has

n = 60

. The lower quartile is estimated at the

15

th value and the upper quartile at the

45

th. Given

Q_1 = 18

and

Q_3 = 34

, find the interquartile range.

●●●●●Level 32 marksStart

A box plot has

Q_1 = 25

and

Q_3 = 45

. An outlier is a value more than

1.5\times\text{IQR}

beyond a quartile. The smallest value in the data is

-8

. Determine whether it is an outlier.

●●●●●Level 32 marksStart

For a cumulative frequency distribution of

200

values, the

90

th percentile is required. At what cumulative frequency is it read, and given the cumulative frequency is

175

x = 60

and

185

x = 70

, which class does it lie in?

●●●●●Level 32 marksStart

A data set has minimum

4

Q_1 = 10

, median

16

Q_3 = 22

and maximum

30

. The box plot is shown.Find the interquartile range and the range.

●●●●●Level 32 marksStart

Two classes sat the same test. Class A has median

58

and IQR

14

; class B has median

62

and IQR

9

. Compare the two distributions.

●●●●●Level 32 marksStart

The stem-and-leaf diagram shows the lengths, in minutes, of the 19 telephone calls made from a small office on one day.

Stem	Leaves
0	4
1
2	3 6 8
3	0 1 4 5 5 7 9
4	1 2 2 6 8
5	0 3 8

Key:

2 \mid 3

●●●●●Level 47 marksStart

Figure 2

Figure 2 is the cumulative frequency graph for the times,

t

minutes, taken by

60

people to complete a task.
(a) Use the graph to estimate the median and the interquartile range. (4 marks)
(b) An outlier is defined as a value more than

1.5 \times

the interquartile range above the upper quartile or below the lower quartile. Show that this data set contains no outliers. (2 marks)

●●●●●Level 46 marksStart

The heights, in cm, of a random sample of plants of variety

A

are summarised below.

\begin{array}{|c|c|c|c|c|}\hline \text{Minimum} & \text{Lower quartile} & \text{Median} & \text{Upper quartile} & \text{Maximum} \\ \hline 8 & 14 & 19 & 25 & 45 \\ \hline \end{array}

An outlier is defined as a value that is more than

1.5 \times

the interquartile range above the upper quartile, or more than

1.5 \times

the interquartile range below the lower quartile.
(a) Show that the maximum height of

45

cm is an outlier. (2 marks)

Figure 2 shows a box plot for the heights of a random sample of plants of variety

B

.Figure 2
(b) Compare the distributions of the heights of the two varieties. State three differences. (3 marks)
(c) The heights of variety

A

are coded using

y = \dfrac{x - 10}{5}

, where

x

cm is the original height.

Find the median and the interquartile range of the coded heights. (3 marks)
(d) State, giving a reason, whether the heights of variety

A

are positively or negatively skewed. (1 mark)

●●●●●Level 49 marksStart

Totals	Morning	Stem	Evening	Totals
(0)		1	8	(1)
(2)	8 6	2	4 7 9	(3)
(6)	9 7 5 5 3 1	3	0 2 6 6 8	(5)
(10)	9 8 6 4 4 4 4 3 2 0	4	1 3 5 5 7 7 9	(7)
(5)	8 6 5 2 1	5	0 2 3 4 6 8	(6)
(3)	7 3 0	6	1 4 6	(3)
(2)	5 2	7	0 3	(2)
(2)	9 3	8	5	(1)
(0)		9	2 6	(2)

Key:

3 \mid 4 \mid 5

	Morning	Evening
Lower quartile	$a$	36
Median	$b$	48
Upper quartile	58	$c$

(b) Find the value of

a

, the value of

b

and the value of

c

(3 marks)
(c) For these morning crossings find, to one decimal place,

(i) the mean number of passengers,

(ii) the standard deviation of the number of passengers.

(You may use

\sum x = 1489

and

\sum x^2 = 81135

where

x

is the number of passengers on a morning crossing.)

(3 marks)

One measure of skewness is found using

\frac{3(\text{mean} - \text{median})}{\text{standard deviation}}

●●●●●Level 410 marksStart

The masses, to the nearest kilogram, of the 36 suitcases checked in for a small charter flight are summarised in the stem and leaf diagram below, where

w

is a constant.

Stem	Leaf	Totals
0	8	(1)
1	1 2 3 4 5 5 6 6 7 8 8 9 9	(13)
2	0 0 1 1 2 2 3 3 4 4 5 6 $w$ $w$ $w$ 9	(16)
3	0 1 2 4 5	(5)
4	7	(1)

Key:

2 \mid 3

w

(3 marks)

The airline calculates the following statistics from these masses

\sum x = 804 \qquad \sum x^2 = 20144

(d) Show that the standard deviation of these masses is 7.80 kg to 3 significant figures.

(2 marks)

For these data an outlier is defined as a value that is greater than

Q_3 + 1.5 \times (Q_3 - Q_1)

or smaller than

Q_1 - 1.5 \times (Q_3 - Q_1)

(e) Show that there is exactly one outlier in these data.

(2 marks)
(f) Describe the skewness of these masses. Give a reason for your answer.

(2 marks)

●●●●●Level 411 marksStart

A music streaming service takes a random sample of 160 songs from one of its playlists and records the length, in seconds, of each song.

The lengths of the songs in the sample are summarised in the following box plot.(a) Use linear interpolation to estimate the probability that a randomly chosen song from this sample is shorter than 232 seconds.

(2 marks)

The service takes the quartiles from this sample and decides to label any song whose length is at least

Q_3 + 1.5 \times (Q_3 - Q_1)

as "extended".
(b) Find the shortest length of a song that the service would label as extended.

(1 mark)

An analyst suggests that the lengths of the songs on the playlist may be modelled by a normal distribution.
(c) Explain whether or not the box plot supports this suggestion.

(1 mark)

The analyst records the length of every song on the playlist and classifies any song whose length is more than 2.5 standard deviations from the mean as unusual.

Assuming that the lengths of the songs on the playlist may be modelled by a normal distribution,
(d) find the probability that a randomly selected song from the playlist would be classified as unusual.

(2 marks)

The mean length of the songs on the playlist is 226 seconds.

Given that a song of length 350 seconds is classified as unusual,
(e) find the maximum possible value of the standard deviation of the lengths of the songs on the playlist.

(2 marks)

●●●●●Level 48 marksStart

k

is a constant.

Totals	Brand A	Stem	Brand B	Totals
(2)	9 4	1	7	(1)
(5)	9 8 7 5 2	2	1 4 6 8	(4)
(10)	8 8 7 6 5 4 3 3 1 0	3	0 1 2 2 3 4 5 5 $k$ 8 9	(11)
(8)	8 7 6 4 3 2 1 0	4	1 3 3 5 6 8	(6)
(4)	6 3 2 0	5	1 3 7	(3)
(1)	1	6	2 6	(2)
(1)	6	7		(0)

Key:

2 \mid 4 \mid 1

means 42 hours for a Brand A phone and 41 hours for a Brand B phone

The median battery life of these Brand B phones is 36 hours.
(a) Find the value of

k

(1 mark)
(b) Find the lower quartile and the upper quartile of the battery life of these Brand A phones.

(2 marks)

The reviewer defines an outlier as an observation that is greater than

Q_3 + 1.5 \times (Q_3 - Q_1)

or less than

Q_1 - 1.5 \times (Q_3 - Q_1)

a

hours, 44 hours and

(2a - 2)

hours.

Given that there is no change to the box plot for the Brand B phones shown in the diagram
(f) find the range of possible values of

a

Show your working clearly.

(3 marks)

●●●●●Level 513 marksStart

Stem	Leaf	Totals
2	3	(1)
3	2 4 5 6 7 8 9 9	(8)
4	0 1 1 2 3 3 4 4 5 6 7 8 9	(13)
5	0 1 2 3 3 5 6 8 9	(9)
6	0 2 4 7	(4)
7		(0)
8	1	(1)
9	4	(1)

Key:

4 \mid 6

means 46 minutes
(a) Find

(i) the value of the median

(ii) the value of

Q_1

and the value of

Q_3

for the journey times during term time.

(3 marks)

For these data an outlier is defined as a value that is greater than

Q_3 + 1.5 \times (Q_3 - Q_1)

or smaller than

Q_1 - 1.5 \times (Q_3 - Q_1)

●●●●●Level 513 marksStart

The thickness,

T

mm, of a paperback book printed by a publisher is normally distributed with

T \sim \mathrm{N}(24, 3^2)

(a) Find the probability that a randomly selected paperback has a thickness of less than 20.4 mm.

(3 marks)
(b) Find

(i) the upper quartile (

Q_3

) of

T

(ii) the lower quartile (

Q_1

) of

T

(4 marks)

A value that is greater than

Q_3 + 1.5 \times (Q_3 - Q_1)

or smaller than

Q_1 - 1.5 \times (Q_3 - Q_1)

is defined as an outlier.
(c) Find these two outlier limits.

(3 marks)

A paperback is selected at random.
(d) Using standardisation, show that the probability that this paperback is an outlier is 0.007 to 3 decimal places.

(3 marks)

Given that this paperback is not an outlier,
(e) showing your working, find the probability that its thickness is greater than 27 mm.

(4 marks)

●●●●●Level 517 marksStart

Histograms and frequency polygons

15 questions

LessonNot started

The histogram shows information about the lengths of time, in minutes, that 160 visitors spent in an art gallery on one day.It is given that 30 visitors spent between 10 and 20 minutes in the gallery.
(a) Find the number of visitors who spent between 40 and 80 minutes in the gallery.

(1 mark)
(b) State the units of the frequency density.

(1 mark)
(c) Suggest one change that the gallery could make to the diagram so that it would provide more information.

(1 mark)
(d) Estimate the number of visitors who spent between 60 and 80 minutes in the gallery.

(1 mark)

●●●●●Level 24 marksStart

A researcher records the time,

t

minutes, that each of

120

customers spends in a shop. The results are shown in the histogram in Figure 1.Figure 1
(a) Find the number of customers who spent less than

10

minutes in the shop. (2 marks)
(b) Estimate the median time spent in the shop, giving your answer to

1

decimal place. (3 marks)
(c) Estimate the number of customers who spent between

15

and

25

minutes in the shop. (3 marks)

●●●●●Level 38 marksStart

A researcher records the time,

t

minutes, that each of

120

customers spends in a shop. The results are shown in the histogram in Figure 1.Figure 1
(a) Estimate the median time spent in the shop, giving your answer to

1

decimal place. (3 marks)
(b) Estimate the number of customers who spent between

15

and

25

minutes in the shop. (3 marks)
(c) Explain why your answers to parts (a) and (b) are only estimates. (1 mark)

●●●●●Level 37 marksStart

The journey times,

t

minutes, of

100

commuters were recorded and summarised in the table.

Time ( $t$ minutes)	$0 \le t < 10$	$10 \le t < 15$	$15 \le t < 20$	$20 \le t < 30$	$30 \le t < 50$
Frequency	$8$	$22$	$30$	$28$	$12$

A histogram is drawn, but the bar for the class

20 \le t < 30

26

minutes. (2 marks)
(d) State one assumption needed for your estimate in part (c). (1 mark)

●●●●●Level 38 marksStart

The distance,

d

km, that an electric scooter travels on a single charge was recorded for each of 120 scooters.

The histogram below represents these data.The table below shows some of the frequencies.

Distance ( $d$ km)	Frequency
$10 \leqslant d < 20$	15
$20 \leqslant d < 25$	20
$25 \leqslant d < 30$
$30 \leqslant d < 40$	35
$40 \leqslant d < 60$

d

(2 marks)
(d) Find an estimate for the mean of

d

(2 marks)

●●●●●Level 38 marksStart

In a histogram, the class

10 \le x < 20

has frequency

30

. Find the frequency density for this class.

●●●●●Level 32 marksStart

On a histogram, a bar for the class

5 \le x < 8

has height (frequency density)

4

. Find the frequency of this class.

●●●●●Level 32 marksStart

The table shows part of a grouped frequency distribution.

\begin{array}{c|ccc} \text{class} & 0\text{–}10 & 10\text{–}25 & 25\text{–}40 \\\\ \hline \text{frequency} & 20 & 45 & 30 \end{array}

A histogram is drawn. Find the frequency density of each class.

●●●●●Level 32 marksStart

On a histogram, the class

20 \le x < 30

is represented by a bar of height

4.5

. Estimate how many values lie in

20 \le x < 24

, assuming they are evenly spread.

●●●●●Level 33 marksStart

In a histogram, a frequency of

10

in the class

0 \le x < 5

is shown by a bar of height

2

. A second class

5 \le x < 15

has frequency

40

. Find the height of the bar for the second class, on the same scale.

●●●●●Level 33 marksStart

A histogram represents

100

observations. The class

30 \le x < 50

has frequency density

1.5

. Find what percentage of the observations lie in this class.

●●●●●Level 33 marksStart

The ages,

a

years, of 120 bicycles brought to a repair workshop are summarised in the table below.

Age ( $a$ years)	Frequency
$0 \leqslant a < 2$	18
$2 \leqslant a < 4$	30
$4 \leqslant a < 5$	24
$5 \leqslant a < 8$	27
$8 \leqslant a < 12$	21

A histogram is drawn to represent these data.

The bar representing the class

2 \leqslant a < 4

is 2.4 cm wide and has an area of 6 cm²
(a) Find the area of the bar representing the class

5 \leqslant a < 8

\frac{Q_3 - 2Q_2 + Q_1}{Q_3 - Q_1}

(d) Evaluate this measure and describe the skewness of these data.

(2 marks)

●●●●●Level 411 marksStart

The lengths,

t

minutes, of 150 calls received by a council helpline on one Monday are summarised in the table below.

Length ( $t$ minutes)	Frequency
$0 \leqslant t < 2$	24
$2 \leqslant t < 5$	45
$5 \leqslant t < 10$	50
$10 \leqslant t < 20$	25
$20 \leqslant t < 30$	6

\sum ft^2 = 12\,762.75

●●●●●Level 412 marksStart

The masses, in grams, of a random sample of 240 potatoes of a certain variety were measured. The results are summarised in the histogram.One of the 240 potatoes is chosen at random, and its mass,

X

grams, is noted.
(a) Show that

\mathrm{P}(160 < X < 180) = 0.183

, correct to 3 significant figures.

(2 marks)

Priya suggests that the distribution of

X

can be well modelled by the distribution

\mathrm{N}(190, 900)

\mathrm{P}(160 < X < 180)

.

(1 mark)

Omar suggests a different model. He uses the midpoints of the classes to calculate estimates,

m

and

s

, for the mean and standard deviation respectively, in grams, of the 240 masses. He then uses the distribution

\mathrm{N}(m, s^2)

as his model.
(d) Use Omar's model to find

\mathrm{P}(160 < X < 180)

.

(4 marks)

The table shows the probabilities of

X

lying in three further ranges, obtained from the histogram and from the two models. Two of the entries,

p

and

q

, are missing.

	$X < 160$	$180 < X < 220$	$X > 220$
Histogram	0.175	0.458	0.183
Priya's model	$p$	0.472	0.159
Omar's model	0.184	$q$	0.197

(e) (i) Find the values of

p

and

q

.

(ii) By considering the different ranges of values of

X

given in the table, together with your answers to parts (a),
(c) and (d), discuss how well the two models fit the original distribution.

(4 marks)

●●●●●Level 514 marksStart

The masses,

w

grams, of 200 strawberries picked on a fruit farm are summarised in the table below.

Mass ( $w$ grams)	Frequency
$5 \leqslant w < 10$	16
$10 \leqslant w < 12$	30
$12 \leqslant w < 14$	52
$14 \leqslant w < 16$	44
$16 \leqslant w < 20$	38
$20 \leqslant w < 30$	20

w

(2 marks)

The lower quartile of

w

is 12.15 to 4 significant figures.
(d) Use linear interpolation to estimate

(i) the upper quartile of

w

(ii) the interquartile range of

w

●●●●●Level 514 marksStart

Correlation and regression

18 questions

LessonNot started

Bea is studying how the depth of snow on a mountain path changes during the spring.

She measures the depth of snow,

d

cm, at time

t

days after 1 March, for each of the

15

days from

t = 0

t = 14

Bea finds the equation of the regression line of

d

t

for her data to be

d = 62.4 - 3.15t

(a) Interpret the gradient of this line.

(1 mark)
(b) Interpret the value

62.4

in Bea's model.

(1 mark)

Bea uses her regression line to estimate the depth of snow on the path when

t = 40

(c) Give a reason why this estimate may not be reliable.

(1 mark)

●●●●●Level 23 marksStart

A transport researcher is investigating whether adults walk more in areas where fewer households own a car. For a random sample of eight local authority areas, the table shows the percentage of households with no car,

x

, and the percentage of adults who walk for at least

20

minutes on three or more days a week,

y

Area	A	B	C	D	E	F	G	H
$x$	$18$	$25$	$31$	$22$	$36$	$28$	$40$	$20$
$y$	$41$	$47$	$55$	$44$	$52$	$46$	$60$	$39$

(a) Calculate the product moment correlation coefficient for these data. (1 mark)

The researcher wishes to test whether there is positive correlation between

x

and

y

for local authority areas in general.

For a sample of size

8

, the critical value of the product moment correlation coefficient at the

5\%

significance level is

0.6215

for a one-tailed test and

0.7067

for a two-tailed test.
(b) Carry out the test at the

5\%

significance level, stating your hypotheses clearly. (4 marks)

The equation of the regression line of

y

x

for these data is

y = 0.863x + 24.3

.
(c) Interpret the value

0.863

in context. (1 mark)
(d) A council officer says: "This shows that not having a car makes people walk more. In an area where

62\%

of households have no car, we can expect about

78\%

of adults to walk regularly." Make two criticisms of this statement. (2 marks)

●●●●●Level 38 marksStart

The product moment correlation coefficient between two variables

x

and

y

r = -0.91

. Describe the correlation.

●●●●●Level 32 marksStart

The product moment correlation coefficient between two variables is

r = 0.05

. A student concludes there is no relationship between them. Comment on this conclusion.

●●●●●Level 32 marksStart

Callum is studying how the depth of water in a small reservoir changed during one summer.

He measures the depth of the water,

y

metres, at time

t

days after 1 June, for

12

values of

t

between

t = 0

and

t = 88

Callum finds the equation of the regression line of

y

t

for his data to be

y = 8.13 - 0.0241t

(a) Interpret the gradient of this line.

(1 mark)

The product moment correlation coefficient between

y

and

t

-0.6602

The critical value for a sample of size

12

at the

5\%

level of significance is

0.4973

(b) Test whether or not there is evidence of a negative correlation between the depth of the water and the time.

You should

state your hypotheses clearly
use a $5\%$ level of significance
state the critical value used

(3 marks)

Nina plots Callum's data on a scatter diagram. The points lie close to a curve: the depth falls steeply at first, reaches its lowest value about

55

days after 1 June, and then rises again.
(c) With reference to Nina's scatter diagram, state, giving a reason, whether or not the regression line

y = 8.13 - 0.0241t

is an appropriate model for these data.

(1 mark)

Nina suggests an improved model using the variable

u = (t - k)^2

, where

k

is a constant.

She obtains the equation

y = 6.10 + 0.00111u

(d) Choose a suitable value for

k

to write Nina's improved model for

y

in terms of

t

only.

(1 mark)

●●●●●Level 46 marksStart

A researcher records the number of hours of revision,

x

, and the test score,

y

, for a random sample of

20

students. The product moment correlation coefficient for the sample is

r = 0.561

.
(a) State what is measured by the product moment correlation coefficient. (1 mark)
(b) Test, at the

1\%

level of significance, whether there is positive correlation between hours of revision and test score.

The critical value of

r

for a one-tailed test at the

1\%

level with

n = 20

0.5368

. (5 marks)
(c) A second researcher analyses the same data but measures the revision time in minutes rather than hours. State, with a reason, the effect this has on the value of

r

. (2 marks)
(d) Explain why this result does not show that revision causes higher test scores. (2 marks)

The equation of the regression line of

y

x

for this sample is

y = 31.2 + 2.4x

.
(e) Use the regression line to estimate the test score of a student who revises for

8

hours, and state one assumption needed for this estimate to be reliable. (2 marks)

●●●●●Level 412 marksStart

A software developer records the number of active users,

N

thousand, of a new app at the end of each of the first

8

weeks after its launch, where

t

is the number of weeks since launch (

t = 1, 2, \ldots, 8

).

The developer believes that the data can be modelled by

N = ab^t

, where

a

and

b

are constants.

The regression line of

y

t

, where

y = \log_{10} N

, is found to be

y = 0.42 + 0.115t

, and the product moment correlation coefficient between

t

and

y

r = 0.964

.
(a) Use the regression line to estimate the values of

a

and

b

, giving your answers to 3 significant figures. (3 marks)
(b) Interpret the value of

b

in this context. (1 mark)
(c) Test, at the

1\%

level of significance, whether there is evidence of positive correlation between

t

and

y

. State your hypotheses clearly, and explain what your conclusion suggests about the developer's model. You may use the critical value

0.7887

. (3 marks)
(d) The developer plans to hire more staff when the number of active users first reaches

50\,000

. Use the model to estimate the value of

t

when this happens, and comment on the reliability of your estimate. (3 marks)

●●●●●Level 410 marksStart

A meteorologist believes that windier days tend to be colder. She records the daily mean windspeed,

w

knots, and the daily mean air temperature,

T\,^\circ

C, on

10

randomly chosen days at a weather station. The results are shown in the table.

\begin{array}{|c|c|c|c|c|c|c|c|c|c|c|}\hline w & 4 & 6 & 7 & 9 & 10 & 12 & 14 & 15 & 18 & 20 \\ \hline T & 16.2 & 13.8 & 14.9 & 17.0 & 13.1 & 14.4 & 11.9 & 15.5 & 10.8 & 12.6 \\ \hline \end{array}

For these data the product moment correlation coefficient is

r = -0.627

and the equation of the regression line of

T

w

T = 16.7 - 0.234w

.
(a) Test, at the

5\%

level of significance, whether these data provide evidence to support the meteorologist's belief. State your hypotheses clearly. (4 marks)
(b) Give an interpretation of the gradient of the regression line in this context. (1 mark)
(c) Use the regression line to estimate the daily mean air temperature on a day when the daily mean windspeed is

12.5

knots, giving your answer to

1

decimal place. (2 marks)
(d) Explain why it would be unreliable to use this regression line to estimate (i) the daily mean air temperature on a day when the daily mean windspeed is

35

knots, (ii) the daily mean windspeed on a day when the daily mean air temperature is

12\,^\circ

C. (2 marks)

●●●●●Level 49 marksStart

A student believes that, in summer, days with a higher maximum temperature tend to have less rainfall. She records the daily maximum temperature,

x\,^\circ

C, and the daily total rainfall,

y

mm, on

8

randomly chosen days in June at a weather station near her school. The results are shown in the table.

\begin{array}{|c|c|c|c|c|c|c|c|c|}\hline x & 16 & 18 & 20 & 21 & 23 & 24 & 26 & 28 \\ \hline y & 2.8 & 5.8 & 4.0 & 4.0 & 1.6 & 1.0 & 3.2 & 0.2 \\ \hline \end{array}

For these data the product moment correlation coefficient is

r = -0.665

, and the equation of the regression line of

y

x

y = 9.46 - 0.302x

.
(a) Test, at the

5\%

level of significance, whether these data support the student's belief. State your hypotheses clearly, and state, with a reason, whether your conclusion would be different at the

1\%

level. (5 marks)
(b) Interpret the gradient of the regression line in this context. (1 mark)
(c) The student uses the regression line to estimate the rainfall on a day in June when the maximum temperature is

21\,^\circ

C, and on a day when it is

34\,^\circ

C. Comment on the reliability of each estimate. (2 marks)

●●●●●Level 48 marksStart

An agricultural researcher is investigating the effect of a fertiliser on tomato plants. She grows tomato plants in

10

plots. Each plot is given a different amount of fertiliser,

f

grams per square metre, and the mean yield,

y

kg per plant, is recorded for each plot. The values of

f

used were between

20

and

80

The researcher summarises the data as follows

\sum f = 465 \qquad \sum y = 45.6 \qquad \sum y^2 = 217.44 \qquad \sum fy = 2295 \qquad S_{ff} = 3552.5

(a) Calculate the exact value of

S_{fy}

and the exact value of

S_{yy}

(3 marks)
(b) Calculate the value of the product moment correlation coefficient between

f

and

y

(2 marks)
(c) Give an interpretation, in context, of your product moment correlation coefficient.

(1 mark)
(d) Show that the equation of the regression line of

y

f

can be written as

y = 2.27 + 0.0491f

45

grams of fertiliser per square metre,

(ii) explain why an estimate of the mean yield per plant for a plot given

150

grams of fertiliser per square metre is not reliable.

(2 marks)

●●●●●Level 412 marksStart

A company asked

11

of its employees for the distance,

d

km, from home to the office and the time,

t

minutes, of their journey to work on one morning. The results are shown in the table below.

Employee	A	B	C	D	E	F	G	H	I	J	K
Distance ( $d$ km)	3	5	6	8	9	11	12	14	17	19	22
Time ( $t$ minutes)	12	15	21	20	95	26	28	31	33	40	43

On that morning, employee E was delayed by a cancelled train.

An outlier is defined as a value that is greater than

Q_3 + 1.5 \times (Q_3 - Q_1)

or smaller than

Q_1 - 1.5 \times (Q_3 - Q_1)

(a) Show that

95

is an outlier for the journey times.

(3 marks)

Leaving out employee E, the company calculated the following summary statistics for the other

10

employees.

\sum d = 117 \qquad \sum t = 269 \qquad S_{dd} = 360.1 \qquad S_{dt} = 572.7

(b) Use these summary statistics to show that the equation of the least squares regression line of

t

d

for these

10

employees is

t = 8.29 + 1.59d

16

km and

30

km from the office.
(d) Using the equation given in part (b), estimate the journey time for

(i) the employee who lives

16

km from the office,

(ii) the employee who lives

30

km from the office.

(3 marks)
(e) State, giving a reason, which of the two estimates found in part (d) would be the more reliable estimate.

(2 marks)

●●●●●Level 412 marksStart

A college records the raw mark,

x

, of each of

12

students on a written chemistry paper and the raw mark,

y

, of each of the same students on a practical chemistry assessment.

Some of the summary statistics are given below.

\sum x = 500 \qquad \sum x^2 = 21\,638 \qquad S_{yy} = 340.92 \qquad S_{xy} = 426.33 \qquad n = 12

(a) (i) Calculate the mean of the

12

raw marks on the written paper.

(ii) Show that the standard deviation of the

12

raw marks on the written paper is

8.19

to 3 significant figures.

(3 marks)
(b) Calculate the product moment correlation coefficient between

x

and

y

(3 marks)

Before publishing the results, the college converts every raw mark into a scaled mark. The formula used is

s = 1.25m + 20

where

m

is the raw mark and

s

is the scaled mark.
(c) Use this formula and the values from part (a) to calculate the mean and the standard deviation of the

12

scaled marks on the written paper. Give your answers to 3 significant figures.

(4 marks)

Given that

$u$ is the scaled mark corresponding to $x$
$v$ is the scaled mark corresponding to $y$

(d) state, giving a reason, the value of the product moment correlation coefficient between

u

and

v

(2 marks)

●●●●●Level 412 marksStart

A farmer records the daily food intake,

x

kg, and the daily milk yield,

y

litres, of each cow in a random sample of

15

of her cows. From the same

15

observations she calculates the following two regression equations.

Regression equation of

y

x

\quad y = 1.5x + 4

Regression equation of

x

y

\quad x = 0.4y + 1.6

(a) (i) Find the point of intersection of these lines.

(ii) Hence show that

\sum x = 120

(4 marks)

Given that

\sum xy = 2130

(b) find

S_{xy}

(4 marks)
(c) Find the product moment correlation coefficient between

x

and

y

(4 marks)

●●●●●Level 412 marksStart

Amara is investigating the relationship between the price of a paperback book and the number of copies of it sold each week in a large chain of bookshops.

She takes a random sample of

15

titles and records, for each title,

its price, $p$ pounds
the number of copies sold in one week, $q$

The points on Amara's scatter diagram lie close to a curve: as

p

increases

q

decreases, steeply at first and then more gently.
(a) Interpret the nature of the relationship between

q

and

p

.

(1 mark)

Amara codes her data using the formulae

x = \log_{10} p \qquad y = \log_{10} q

The product moment correlation coefficient between

x

and

y

-0.8463

The critical value for a sample of size

15

at the

5\%

level of significance is

0.4409

(b) Test whether or not there is significant evidence of a negative correlation between

x

and

y

.

You should

state your hypotheses clearly
use a $5\%$ level of significance
state the critical value used

(3 marks)

The equation of the line of best fit of

y

x

y = 3.4 - 1.6x

(c) Use the equation of the line of best fit of

y

x

to find a model for

q

in terms of

p

in the form

q = ap^{k}

where

a

and

k

are constants to be found. Give the value of

a

to 3 significant figures.

(5 marks)

●●●●●Level 59 marksStart

Each week, the owner of a garden centre records the number,

c

, of bags of compost sold and the amount, £

a

, spent on local radio advertising during that week. The owner wants to test whether there is linear correlation between

c

and

a

.

The owner chooses a random sample of 22 weeks and finds that, for these weeks, the product-moment correlation coefficient between

c

and

a

r = 0.583

(3 sf).
(a) Using the table below, carry out the test at the 1% significance level.

(5 marks)
(b) Describe briefly two main features of a scatter diagram of the 22 pairs of values of

c

and

a

. You do not need to draw the diagram.

(2 marks)
(c) The owner makes the following statement.

“The value of

r

shows that spending more on radio advertising makes customers buy more compost. So we should double the amount we spend on radio advertising.”

Comment briefly on this statement, making reference to the context.

(2 marks)
(d) The table below includes the value 0.4921. Give a statement about a probability to explain the meaning of this value.

Critical values of the product-moment correlation coefficient

	1-tail test: 5%	2.5%	1%	0.5%
	2-tail test: 10%	5%	2%	1%
$n = 20$	0.3783	0.4438	0.5155	0.5614
$n = 21$	0.3687	0.4329	0.5034	0.5487
$n = 22$	0.3598	0.4227	0.4921	0.5368
$n = 23$	0.3515	0.4132	0.4815	0.5256

(2 marks)

●●●●●Level 511 marksStart

An auction house sells used cars. Before each sale, the auction house gives each car a guide price, £

x

thousand. For a random sample of

10

cars, the guide price and the sale price, £

y

thousand, were recorded. The guide prices of these cars were between £

4

thousand and £

31

thousand.

The data are summarised as follows

\sum x = 150 \qquad \sum y = 134.7 \qquad \sum y^2 = 2242.73 \qquad \sum xy = 2584.4 \qquad S_{xx} = 746

(a) Calculate the exact value of

S_{xy}

and the exact value of

S_{yy}

(3 marks)
(b) Calculate the value of the product moment correlation coefficient between

x

and

y

(2 marks)
(c) Give an interpretation of your product moment correlation coefficient.

(1 mark)
(d) Show that the equation of the regression line of

y

x

can be written as

y = 2.13 + 0.756x

10\,000

(ii) explain why an estimate of the sale price of a car with a guide price of £

60\,000

is not reliable.

(2 marks)

The auction house says that a car has sold "below guide" if its sale price is less than

90\%

of its guide price.
(g) Using the equation of the regression line given in part (d), find the range of values of

x

for which a car is expected to sell below guide.

(2 marks)

●●●●●Level 514 marksStart

s

, and the amount of electricity generated by the panels that day,

g

kWh.

Kofi takes a random sample of

12

days and codes the amount of electricity generated so that

w = 20g

These data are summarised as follows

S_{ss} = 140.8425 \qquad S_{sw} = 1881.35 \qquad \sum s = 79.5 \qquad \sum w = 1570 \qquad \sum w^2 = 232\,756

(b) Show that

S_{ww} = 27\,347.67

to 2 decimal places.

(1 mark)
(c) Find the value of

S_{sg}

and the value of

S_{gg}

(3 marks)
(d) Find the product moment correlation coefficient between

g

and

s

(2 marks)
(e) Give an interpretation, in context, of your product moment correlation coefficient.

(1 mark)
(f) Show that the equation of the regression line of

w

s

w = 42.3 + 13.4s

where the values of the intercept and the gradient are given to 3 significant figures.

(3 marks)
(g) Write down an equation of the regression line of

g

s

(1 mark)
(h) Using your equation in part (g)

(i) estimate the amount of electricity generated on a day with

7

hours of sunshine,

(ii) interpret the effect that an increase of

1

hour of sunshine is expected to have on the amount of electricity generated.

(2 marks)

●●●●●Level 514 marksStart

A garden centre is investigating the growth of a variety of sunflower. For each of

12

seedlings, the number of days after planting,

x

, and the height of the seedling,

y

cm, are recorded, where

5 \leqslant x \leqslant 40

The regression line of

y

x

is found to be

y = -6.7 + 1.1x

(a) Give an interpretation of the gradient of the regression line.

(1 mark)
(b) Use the regression line to estimate the height of a seedling

55

days after planting.

(2 marks)
(c) Comment on the reliability of your estimate, giving a reason for your answer.

(2 marks)

Using the regression line of

y

x

and the following summary statistics

\sum y = 212.2 \qquad \sum y^2 = 5606.62 \qquad \sum x^2 = 7378 \qquad n = 12

(d) show that the product moment correlation coefficient for these data is

0.983

x

and

y

.

(2 marks)

●●●●●Level 514 marksStart

Exponential models and regression

17 questions

LessonNot started

The regression line of

\log_{10} y

x

\log_{10} y = 0.3 + 0.2x

. Express

y

in terms of

x

, giving the constants to 3 significant figures.

●●●●●Level 22 marksStart

Using the model

\log_{10} y = 0.3 + 0.2x

, estimate the value of

y

when

x = 5

●●●●●Level 22 marksStart

The regression line of

\log_{10} y

x

\log_{10} y = 0.5 + 0.3x

.

Express the model in the form

y = ab^{x}

, giving

a

and

b

to 3 significant figures. (3 marks)

●●●●●Level 23 marksStart

The regression line of

\log_{10} y

\log_{10} x

\log_{10} y = 0.2 + 1.5\log_{10} x

.

Express the model in the form

y = ax^{n}

, giving

a

to 3 significant figures. (4 marks)

●●●●●Level 34 marksStart

A population

P

t

years after monitoring began, is modelled by the regression line

\log_{10} P = 2 + 0.05t

(a) Find the population when monitoring began. (2 marks)
(b) Estimate the population after

10

years, to 3 significant figures. (2 marks)

●●●●●Level 34 marksStart

The value

V

(in £) of a machine

t

years after purchase is modelled by

\log_{10} V = 3 - 0.1t

Find the time at which the model predicts the value will have halved from its purchase price, giving your answer to 3 significant figures. (4 marks)

●●●●●Level 34 marksStart

A scientist plots

\log_{10} y

against

x

for a data set and finds the points lie close to a straight line.

Explain why this supports a model of the form

y = ab^{x}

, and state what the gradient and intercept of the line estimate. (3 marks)

●●●●●Level 33 marksStart

For a data set, the regression line of

\log_{10} y

x

passes through the points

(0, 1.2)

and

(5, 2.2)

.

Find the model in the form

y = ab^{x}

, giving

a

and

b

to 3 significant figures. (5 marks)

●●●●●Level 35 marksStart

The regression line of

\log_{10} y

\log_{10} x

for a data set is

\log_{10} y = 0.301 + 2\log_{10} x

(a) Express the model in the form

y = ax^{n}

, giving

a

to 3 significant figures. (3 marks)
(b) Use the model to estimate

y

when

x = 5

. (1 mark)

●●●●●Level 34 marksStart

The regression line of

\log_{10} y

x

\log_{10} y = 0.6 + 0.4x

.
(a) Express

y

in the form

y = ab^x

, giving

a

and

b

to 3 s.f.
(b) Hence estimate

y

when

x = 3

●●●●●Level 34 marksStart

The regression line of

\log_{10} y

\log_{10} x

\log_{10} y = 0.5 + 1.5\log_{10} x

. Express

y

in the form

y = ax^n

, giving the constants to 3 s.f.

●●●●●Level 32 marksStart

A population

P

after

t

years is modelled by the regression line

\log_{10} P = 2 + 0.05t

.
(a) Express

P

in the form

P = ab^t

, giving the constants to 3 s.f.
(b) State the initial population.

●●●●●Level 33 marksStart

The regression line of

\log_{10} y

x

has gradient

0.15

and intercept

0.8

.
(a) Find

a

and

b

so that

y = ab^x

, to 3 s.f.
(b) Estimate

y

when

x = 10

●●●●●Level 34 marksStart

The braking distance

d

of a car at speed

t

is modelled by

\log_{10} d = 0.7 + 0.5\log_{10} t

.
(a) Show that

d = a\,t^{n}

, stating

a

(3 s.f.) and

n

.
(b) Estimate

d

when

t = 4

●●●●●Level 34 marksStart

The value

V

of a machine after

t

years is modelled by

\log_{10} V = 1.2 - 0.1t

.
(a) Express

V

in the form

V = ab^t

, to 3 s.f.
(b) Interpret the value of

b

●●●●●Level 33 marksStart

A software developer records the number of active users,

N

thousand, of a new app at the end of each of the first

8

weeks after its launch, where

t

is the number of weeks since launch (

t = 1, 2, \ldots, 8

).

The developer believes that the data can be modelled by

N = ab^t

, where

a

and

b

are constants.

The regression line of

y

t

, where

y = \log_{10} N

, is found to be

y = 0.42 + 0.115t

, and the product moment correlation coefficient between

t

and

y

r = 0.964

.
(a) Use the regression line to estimate the values of

a

and

b

, giving your answers to 3 significant figures. (3 marks)
(b) Interpret the value of

b

in this context. (1 mark)
(c) Test, at the

1\%

level of significance, whether there is evidence of positive correlation between

t

and

y

. State your hypotheses clearly, and explain what your conclusion suggests about the developer's model. You may use the critical value

0.7887

. (3 marks)
(d) The developer plans to hire more staff when the number of active users first reaches

50\,000

. Use the model to estimate the value of

t

when this happens, and comment on the reliability of your estimate. (3 marks)

●●●●●Level 410 marksStart

Femi is investigating how the membership of a new climbing club has grown.

At the end of each of the first

10

months after the club opened he records

the time, $t$ months, since the club opened
the number of members, $m$ , of the club

The points on Femi's scatter diagram lie close to a curve: as

t

increases

m

increases, slowly at first and then more and more quickly.
(a) Interpret the nature of the relationship between

m

and

t

.

(1 mark)

Femi codes his data using the formula

y = \log_{10} m

The product moment correlation coefficient between

t

and

y

0.9317

The critical value for a sample of size

10

at the

5\%

level of significance is

0.5494

(b) Test whether or not there is significant evidence of a positive correlation between

t

and

y

.

You should

state your hypotheses clearly
use a $5\%$ level of significance
state the critical value used

(3 marks)

The equation of the line of best fit of

y

t

y = 1.62 + 0.058t

(c) Use the equation of the line of best fit of

y

t

to find a model for

m

in terms of

t

in the form

m = ab^{t}

where

a

and

b

are constants to be found. Give the value of

a

and the value of

b

to 3 significant figures.

(5 marks)

●●●●●Level 59 marksStart

Measuring correlation (the PMCC)

29 questions

LessonNot started

For a set of bivariate data,

S_{xy} = 100

S_{xx} = 72

and

S_{yy} = 152

. Find the product moment correlation coefficient

r

, to 3 decimal places.

●●●●●Level 22 marksStart

For a set of bivariate data,

S_{xy} = 60

S_{xx} = 90

and

S_{yy} = 160

.

Calculate the product moment correlation coefficient. (2 marks)

●●●●●Level 22 marksStart

For each of the following, describe the correlation and suggest what it means in context.
(a) Hours of revision and exam mark:

r = 0.92

. (2 marks)
(b) Shoe size and reaction time:

r = -0.15

. (1 mark)

●●●●●Level 23 marksStart

A cafe owner records the maximum daily temperature,

x

°C, and her daily ice-cream sales,

y

(units sold), on

12

randomly chosen days. The temperatures range from

12

°C to

28

°C. The product moment correlation coefficient for the sample is

r = 0.716

.
(a) State what the product moment correlation coefficient measures, and interpret the value

0.716

in this context. (2 marks)
(b) Test, at the

5\%

level of significance, whether there is evidence of positive correlation between temperature and ice-cream sales. State your hypotheses clearly. You may use the critical value

0.4973

. (4 marks)
(c) The owner uses her regression line to predict sales on a day when the temperature is

35

°C, and concludes that hot weather causes people to buy ice cream. Comment on each of these. (2 marks)

●●●●●Level 38 marksStart

x

, and the percentage of adults who walk for at least

20

minutes on three or more days a week,

y

Area	A	B	C	D	E	F	G	H
$x$	$18$	$25$	$31$	$22$	$36$	$28$	$40$	$20$
$y$	$41$	$47$	$55$	$44$	$52$	$46$	$60$	$39$

(a) Calculate the product moment correlation coefficient for these data. (1 mark)

The researcher wishes to test whether there is positive correlation between

x

and

y

for local authority areas in general.

For a sample of size

8

, the critical value of the product moment correlation coefficient at the

5\%

significance level is

0.6215

for a one-tailed test and

0.7067

for a two-tailed test.
(b) Carry out the test at the

5\%

significance level, stating your hypotheses clearly. (4 marks)

The equation of the regression line of

y

x

for these data is

y = 0.863x + 24.3

.
(c) Interpret the value

0.863

in context. (1 mark)
(d) A council officer says: "This shows that not having a car makes people walk more. In an area where

62\%

of households have no car, we can expect about

78\%

of adults to walk regularly." Make two criticisms of this statement. (2 marks)

●●●●●Level 38 marksStart

For

8

pairs of data:

\sum x = 48

\sum y = 96

\sum xy = 676

\sum x^2 = 360

\sum y^2 = 1304

. Find

r

to 3 d.p.

●●●●●Level 34 marksStart

For

10

pairs of data:

\sum x = 50

\sum y = 100

\sum xy = 590

\sum x^2 = 300

\sum y^2 = 1200

. Find

r

to 3 d.p.

●●●●●Level 34 marksStart

The product moment correlation coefficient between daily temperature and ice-cream sales is found to be

r = 0.88

. Interpret this value in context.

●●●●●Level 32 marksStart

A student codes the variable

x

using

p = \dfrac{x - 50}{5}

and finds the PMCC between

p

and

y

to be

0.74

. State, with a reason, the PMCC between

x

and

y

●●●●●Level 32 marksStart

The table shows five pairs of data.

\begin{array}{c|ccccc} x & 1 & 2 & 3 & 4 & 5 \\\\ \hline y & 2 & 3 & 5 & 4 & 6 \end{array}

Calculate the product moment correlation coefficient, to 3 d.p.

●●●●●Level 34 marksStart

For

12

pairs of data:

\sum x = 300

\sum y = 600

\sum xy = 15800

\sum x^2 = 8200

\sum y^2 = 31000

. Find

r

to 3 d.p. and describe the correlation.

●●●●●Level 35 marksStart

For

6

pairs of data:

\sum x = 30

\sum y = 60

\sum xy = 370

\sum x^2 = 200

\sum y^2 = 760

. Find

r

to 3 d.p. and comment on whether a linear model is appropriate.

●●●●●Level 35 marksStart

For

10

pairs of data:

\sum x = 60

\sum y = 80

\sum xy = 520

\sum x^{2} = 400

\sum y^{2} = 740

.

Calculate the product moment correlation coefficient, giving your answer to 3 significant figures. (5 marks)

●●●●●Level 35 marksStart

A student codes data using

p = \dfrac{x - 100}{10}

and

q = 2y

, and calculates the product moment correlation coefficient of the coded data as

r_{pq} = 0.78

.

Write down the product moment correlation coefficient between

x

and

y

, justifying your answer. (3 marks)

●●●●●Level 33 marksStart

The table shows five pairs of data.

\begin{array}{c|ccccc} x & 1 & 2 & 3 & 4 & 5 \\ \hline y & 2 & 3 & 5 & 6 & 9 \end{array}

Calculate the product moment correlation coefficient, giving your answer to 3 significant figures. (5 marks)

●●●●●Level 35 marksStart

A student calculates summary statistics for a data set and obtains

S_{xy} = 50

S_{xx} = 20

S_{yy} = 80

.

Show that these values must contain an error. (3 marks)

●●●●●Level 33 marksStart

The correlation between daily ice-cream sales and daily drowning incidents at a seaside town over a year is

r = 0.85

.

A reporter claims this shows that ice cream causes drowning. Explain what is wrong with this claim, and suggest a more plausible explanation. (3 marks)

●●●●●Level 33 marksStart

A motoring journalist plans to carry out a statistical investigation to test whether there is linear correlation between the engine size,

E

litres, and the fuel economy,

F

miles per gallon, of new family cars.
(a) Explain why a 1-tail test is appropriate in this context.

(1 mark)

The journalist records the values of

E

and

F

for a random sample of 16 new family cars. She calculates the product-moment correlation coefficient for the sample and finds that the value is

-0.462

.
(b) Use the table below to carry out the test at the 5% significance level.

Critical values of the product-moment correlation coefficient

	1-tail test: 5%	2.5%	1%	0.5%
	2-tail test: 10%	5%	2%	1%
$n = 14$	0.4575	0.5324	0.6120	0.6614
$n = 15$	0.4409	0.5140	0.5923	0.6411
$n = 16$	0.4259	0.4973	0.5742	0.6226
$n = 17$	0.4124	0.4821	0.5577	0.6055

(5 marks)

●●●●●Level 46 marksStart

A researcher records the number of hours of revision,

x

, and the test score,

y

, for a random sample of

20

students. The product moment correlation coefficient for the sample is

r = 0.561

.
(a) State what is measured by the product moment correlation coefficient. (1 mark)
(b) Test, at the

1\%

level of significance, whether there is positive correlation between hours of revision and test score.

The critical value of

r

for a one-tailed test at the

1\%

level with

n = 20

0.5368

. (5 marks)
(c) A second researcher analyses the same data but measures the revision time in minutes rather than hours. State, with a reason, the effect this has on the value of

r

. (2 marks)
(d) Explain why this result does not show that revision causes higher test scores. (2 marks)

The equation of the regression line of

y

x

for this sample is

y = 31.2 + 2.4x

.
(e) Use the regression line to estimate the test score of a student who revises for

8

hours, and state one assumption needed for this estimate to be reliable. (2 marks)

●●●●●Level 412 marksStart

A meteorologist believes that windier days tend to be colder. She records the daily mean windspeed,

w

knots, and the daily mean air temperature,

T\,^\circ

C, on

10

randomly chosen days at a weather station. The results are shown in the table.

\begin{array}{|c|c|c|c|c|c|c|c|c|c|c|}\hline w & 4 & 6 & 7 & 9 & 10 & 12 & 14 & 15 & 18 & 20 \\ \hline T & 16.2 & 13.8 & 14.9 & 17.0 & 13.1 & 14.4 & 11.9 & 15.5 & 10.8 & 12.6 \\ \hline \end{array}

For these data the product moment correlation coefficient is

r = -0.627

and the equation of the regression line of

T

w

T = 16.7 - 0.234w

.
(a) Test, at the

5\%

12.5

knots, giving your answer to

1

decimal place. (2 marks)
(d) Explain why it would be unreliable to use this regression line to estimate (i) the daily mean air temperature on a day when the daily mean windspeed is

35

knots, (ii) the daily mean windspeed on a day when the daily mean air temperature is

12\,^\circ

C. (2 marks)

●●●●●Level 49 marksStart

A student believes that, in summer, days with a higher maximum temperature tend to have less rainfall. She records the daily maximum temperature,

x\,^\circ

C, and the daily total rainfall,

y

mm, on

8

randomly chosen days in June at a weather station near her school. The results are shown in the table.

\begin{array}{|c|c|c|c|c|c|c|c|c|}\hline x & 16 & 18 & 20 & 21 & 23 & 24 & 26 & 28 \\ \hline y & 2.8 & 5.8 & 4.0 & 4.0 & 1.6 & 1.0 & 3.2 & 0.2 \\ \hline \end{array}

For these data the product moment correlation coefficient is

r = -0.665

, and the equation of the regression line of

y

x

y = 9.46 - 0.302x

.
(a) Test, at the

5\%

level of significance, whether these data support the student's belief. State your hypotheses clearly, and state, with a reason, whether your conclusion would be different at the

1\%

21\,^\circ

C, and on a day when it is

34\,^\circ

C. Comment on the reliability of each estimate. (2 marks)

●●●●●Level 48 marksStart

An agricultural researcher is investigating the effect of a fertiliser on tomato plants. She grows tomato plants in

10

plots. Each plot is given a different amount of fertiliser,

f

grams per square metre, and the mean yield,

y

kg per plant, is recorded for each plot. The values of

f

used were between

20

and

80

The researcher summarises the data as follows

\sum f = 465 \qquad \sum y = 45.6 \qquad \sum y^2 = 217.44 \qquad \sum fy = 2295 \qquad S_{ff} = 3552.5

(a) Calculate the exact value of

S_{fy}

and the exact value of

S_{yy}

(3 marks)
(b) Calculate the value of the product moment correlation coefficient between

f

and

y

(2 marks)
(c) Give an interpretation, in context, of your product moment correlation coefficient.

(1 mark)
(d) Show that the equation of the regression line of

y

f

can be written as

y = 2.27 + 0.0491f

45

grams of fertiliser per square metre,

(ii) explain why an estimate of the mean yield per plant for a plot given

150

grams of fertiliser per square metre is not reliable.

(2 marks)

●●●●●Level 412 marksStart

A company asked

11

of its employees for the distance,

d

km, from home to the office and the time,

t

minutes, of their journey to work on one morning. The results are shown in the table below.

Employee	A	B	C	D	E	F	G	H	I	J	K
Distance ( $d$ km)	3	5	6	8	9	11	12	14	17	19	22
Time ( $t$ minutes)	12	15	21	20	95	26	28	31	33	40	43

On that morning, employee E was delayed by a cancelled train.

An outlier is defined as a value that is greater than

Q_3 + 1.5 \times (Q_3 - Q_1)

or smaller than

Q_1 - 1.5 \times (Q_3 - Q_1)

(a) Show that

95

is an outlier for the journey times.

(3 marks)

Leaving out employee E, the company calculated the following summary statistics for the other

10

employees.

\sum d = 117 \qquad \sum t = 269 \qquad S_{dd} = 360.1 \qquad S_{dt} = 572.7

(b) Use these summary statistics to show that the equation of the least squares regression line of

t

d

for these

10

employees is

t = 8.29 + 1.59d

16

km and

30

km from the office.
(d) Using the equation given in part (b), estimate the journey time for

(i) the employee who lives

16

km from the office,

(ii) the employee who lives

30

km from the office.

(3 marks)
(e) State, giving a reason, which of the two estimates found in part (d) would be the more reliable estimate.

(2 marks)

●●●●●Level 412 marksStart

A college records the raw mark,

x

, of each of

12

students on a written chemistry paper and the raw mark,

y

, of each of the same students on a practical chemistry assessment.

Some of the summary statistics are given below.

\sum x = 500 \qquad \sum x^2 = 21\,638 \qquad S_{yy} = 340.92 \qquad S_{xy} = 426.33 \qquad n = 12

(a) (i) Calculate the mean of the

12

raw marks on the written paper.

(ii) Show that the standard deviation of the

12

raw marks on the written paper is

8.19

to 3 significant figures.

(3 marks)
(b) Calculate the product moment correlation coefficient between

x

and

y

(3 marks)

Before publishing the results, the college converts every raw mark into a scaled mark. The formula used is

s = 1.25m + 20

where

m

is the raw mark and

s

is the scaled mark.
(c) Use this formula and the values from part (a) to calculate the mean and the standard deviation of the

12

scaled marks on the written paper. Give your answers to 3 significant figures.

(4 marks)

Given that

$u$ is the scaled mark corresponding to $x$
$v$ is the scaled mark corresponding to $y$

(d) state, giving a reason, the value of the product moment correlation coefficient between

u

and

v

(2 marks)

●●●●●Level 412 marksStart

A farmer records the daily food intake,

x

kg, and the daily milk yield,

y

litres, of each cow in a random sample of

15

of her cows. From the same

15

observations she calculates the following two regression equations.

Regression equation of

y

x

\quad y = 1.5x + 4

Regression equation of

x

y

\quad x = 0.4y + 1.6

(a) (i) Find the point of intersection of these lines.

(ii) Hence show that

\sum x = 120

(4 marks)

Given that

\sum xy = 2130

(b) find

S_{xy}

(4 marks)
(c) Find the product moment correlation coefficient between

x

and

y

(4 marks)

●●●●●Level 412 marksStart

Each week, the owner of a garden centre records the number,

c

, of bags of compost sold and the amount, £

a

, spent on local radio advertising during that week. The owner wants to test whether there is linear correlation between

c

and

a

.

The owner chooses a random sample of 22 weeks and finds that, for these weeks, the product-moment correlation coefficient between

c

and

a

r = 0.583

(3 sf).
(a) Using the table below, carry out the test at the 1% significance level.

(5 marks)
(b) Describe briefly two main features of a scatter diagram of the 22 pairs of values of

c

and

a

. You do not need to draw the diagram.

(2 marks)
(c) The owner makes the following statement.

“The value of

r

	1-tail test: 5%	2.5%	1%	0.5%
	2-tail test: 10%	5%	2%	1%
$n = 20$	0.3783	0.4438	0.5155	0.5614
$n = 21$	0.3687	0.4329	0.5034	0.5487
$n = 22$	0.3598	0.4227	0.4921	0.5368
$n = 23$	0.3515	0.4132	0.4815	0.5256

(2 marks)