Figure — understand the maths, don't just answer it

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

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

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

on

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

on

s

is

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

on

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

on

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

on

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

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 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 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

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

Correlation and regression

18 questions

LessonNot started

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

is

-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

on

x

is

y = 3.4 - 1.6x

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

y

on

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

is

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

on

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

on

s

is

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

on

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

on

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

on

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

Exponential models and regression

17 questions

LessonNot started

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

is

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

on

t

is

y = 1.62 + 0.058t

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

y

on

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

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

is

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)