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

Measures of spread, standard deviation and coding

38 questions

LessonNot started

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

Cumulative frequency, box plots and outliers

18 questions

LessonNot started

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

Histograms and frequency polygons

15 questions

LessonNot started

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

Correlation and regression

18 questions

LessonNot started

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

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 mean yield per plant for a plot given

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

Exponential models and regression

17 questions

LessonNot started

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

Measuring correlation (the PMCC)

29 questions

LessonNot started

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)