Browse Hypothesis testing

a null hypothesis corresponding to no change in the proportion of households that put out garden waste for recycling
an alternative hypothesis corresponding to Yusuf's belief

(1 mark)
(b) Using a

5\%

level of significance, find the critical region for a two-tailed test of the null hypothesis in part (a).

You should state the probability of rejection in each tail, which should be less than

0.025

(3 marks)
(c) Find the actual level of significance of the test based on your critical region from part (b).

(1 mark)

Yusuf observes that

17

of the

40

households put out garden waste for recycling.
(d) With reference to part (b), comment on Yusuf's belief.

(1 mark)

●●●●●Level 46 marksStart

A study was made of the loaves of bread baked at the Millgate bakery. It was found that the masses of these loaves were normally distributed with a mean of

802

g and a standard deviation of

12

g.
(a) Find the proportion of these loaves with a mass of more than

820

g.

(1 mark)

A student claimed that the mean mass of the loaves baked at the Riverbank bakery was different from the mean mass of the loaves baked at the Millgate bakery.

A random sample of

30

loaves baked at the Riverbank bakery had a mean mass of

796.5

g

The student assumed that the standard deviation of the masses of the loaves was

12

g both for the Millgate bakery and for the Riverbank bakery.
(b) Use a suitable test to assess the student's claim.

You should

state your hypotheses clearly
use a $5\%$ level of significance

(4 marks)
(c) Find the p-value for the test in part (b).

(1 mark)

●●●●●Level 46 marksStart

A study was made of the times taken by the trainees at the Northgate depot to complete a standard assembly task. It was found that these times were normally distributed with a mean of

46.0

minutes and a standard deviation of

5.2

minutes.
(a) Find the proportion of these trainees who complete the task in less than

40

minutes.

(1 mark)

A manager claimed that the mean time taken by the trainees at the Riverside depot was different from the mean time taken by the trainees at the Northgate depot.

A random sample of

20

trainees at the Riverside depot had a mean time of

44.2

minutes

The manager assumed that the standard deviation of the times was

5.2

minutes for the trainees at both depots.
(b) Use a suitable test to assess the manager's claim.

You should

state your hypotheses clearly
use a $5\%$ level of significance

(4 marks)
(c) Find the p-value for the test in part (b).

(1 mark)

●●●●●Level 46 marksStart

A game uses a spinner which is divided into five equal sectors, one of which is coloured red. Kofi suspects that the spinner is biased against red, so that when it is spun, it is less likely to land on red than if it were fair.

In order to test this suspicion, Kofi plans to spin the spinner 90 times and note the number of times,

X

, that it lands on red. He will then carry out a hypothesis test at the 5% significance level.
(a) State appropriate null and alternative hypotheses for the test.

(2 marks)
(b) Determine the largest value of

X

that would provide evidence at the 5% significance level that the spinner is biased against red.

(3 marks)

Later, Kofi's sister carries out a similar test, at the 5% significance level. She also spins the spinner 90 times.
(c) It is given that the spinner is fair.

Find the probability that the conclusion of the test is that there is significant evidence that the spinner is biased against red.

(1 mark)

●●●●●Level 46 marksStart

Gwen is a driving instructor. She claims that 75% of her pupils pass their practical driving test at their first attempt. Tomasz suspects that Gwen is over-confident about the first-attempt pass rate amongst her pupils. Tomasz contacts a random sample of 24 of Gwen's former pupils, and finds that 13 of them passed at their first attempt. He then performs a hypothesis test.

Carry out the test at the 5% significance level.

(7 marks)

●●●●●Level 47 marksStart

In a football league, the number of goals scored in a match is modelled by a Poisson distribution with mean

2.7

. The numbers of goals scored in different matches are independent. Each round of the league consists of

10

matches.
(a) Find the probability that more than

4

goals are scored in a randomly selected match.

(2 marks)

The league introduces a new rule to encourage attacking play. The total number of goals,

T

, scored in the next round of

10

matches will be used to test whether the mean number of goals per match has increased.
(b) Write down suitable null and alternative hypotheses for this test.

(1 mark)
(c) Using a suitable approximation, show that the critical region for this test, at the

5\%

level of significance, is

T \geqslant 37

(5 marks)
(d) Using the same approximation, find the actual significance level of this test.

(2 marks)

In the next round,

35

goals are scored.
(e) Comment on this result in the light of your critical region.

(2 marks)

●●●●●Level 412 marksStart

A wildlife hospital admits injured hedgehogs. During the summer, hedgehogs are admitted at random, at a mean rate of

3.5

per day.
(a) State one assumption, in context, that is needed for the number of hedgehogs admitted in a day to be modelled by a Poisson distribution.

(1 mark)
(b) Use a suitable approximation to show that the probability of fewer than

30

admissions in a

10

-day period is

0.18

, correct to

2

decimal places.

Show your working clearly.

(Solutions relying on calculator technology are not acceptable.)

(5 marks)

In one

10

-day period there are fewer than

30

admissions.
(c) Using the same approximation and your answer to part (b), find the probability that there are fewer than

25

admissions in this period.

(3 marks)

During a heatwave,

49

hedgehogs are admitted in a

10

-day period.
(d) Using a suitable approximation, test, at the

1\%

level of significance, whether or not there is evidence that the mean rate of admissions has increased. State your hypotheses clearly.

(7 marks)

●●●●●Level 516 marksStart

Potholes form on the roads of a city at random, at a mean rate of

0.8

per km of road each winter.
(a) Find the probability that more than

2

potholes form on a

1.5

km stretch of road in one winter.

(2 marks)

The council selects a length of road,

x

km long, for a survey. When a normal approximation is used, the probability that at least

88

potholes form on this length of road in one winter is

0.8944

(b) Show that

x = 125

(7 marks)

The council resurfaces these

125

km of road. In the following winter,

82

potholes form on them.
(c) Using a suitable approximation, test, at the

5\%

level of significance, whether or not there is evidence that the mean rate at which potholes form on these roads has decreased. State your hypotheses clearly.

(7 marks)

●●●●●Level 516 marksStart

During weekday evenings, requests for rides arrive at a taxi company's app at random, at a mean rate of

r

per minute.

When a normal approximation is used, the probability of at least

60

requests in a

20

-minute period is

0.0668

(a) Show that

r = 2.45

(7 marks)

Of the requests,

20\%

are for journeys to the airport, independently of each other.
(b) Find the probability that there are fewer than

2

requests for journeys to the airport in a

5

-minute period.

(3 marks)

The company then raises its evening fares. In a randomly selected hour after the rise, there are

128

requests.
(c) Using a suitable approximation, test, at the

5\%

level of significance, whether or not there is evidence that the mean rate of requests has changed. State your hypotheses clearly.

(7 marks)

●●●●●Level 517 marksStart

A city bike-hire scheme charges

£2

for a short ride,

£4

for a medium ride and

£7

for a long ride. The operator models the rides taken as short, medium and long in the ratio

2 : 1 : 1

respectively.

A random sample of

2

rides is taken from the very large number of rides taken. The random variable

R

represents the range, in pounds, of the charges for the

2

rides.
(a) Show that, according to the model,

P(R = 5) = \frac14

(2 marks)
(b) Find the sampling distribution of

R

(4 marks)
(c) Find

E(R)

(2 marks)

The operator suspects that samples of

2

rides with a range of

£5

occur more often than the model suggests. She takes

20

random samples of

2

rides and finds that

9

of these samples have a range of

£5

(d) Test, at the

5\%

level of significance, whether or not there is evidence to support the operator's suspicion. State your hypotheses clearly.

(5 marks)

●●●●●Level 513 marksStart

Critical regions and actual significance level

28 questions

LessonNot started

The random variable

Y

has a Poisson distribution with mean

\lambda

. A single observed value of

Y

is to be used to test

H_0: \lambda = 3.5

against

H_1: \lambda < 3.5

, using a

5\%

level of significance.
(a) Find the critical region for this test.

(2 marks)
(b) Write down the actual significance level of this test.

(1 mark)

●●●●●Level 23 marksStart

In a large city,

45\%

of journeys to work are made by car.

Priya believes that the proportion of journeys to work made by car by people who live in the city centre is different from

45\%

She takes a random sample of

20

people who live in the city centre and records the number,

X

, who travel to work by car.
(a) Write down

a null hypothesis corresponding to no change in the proportion
an alternative hypothesis corresponding to Priya's belief

(1 mark)
(b) Using a

5\%

level of significance, find the critical region for a two-tailed test of the null hypothesis in part (a).

You should state the probability of rejection in each tail, which should be less than

0.025

(3 marks)

●●●●●Level 34 marksStart

●●●●●Level 36 marksStart

For a test of

H_0: p = 0.25

against

H_1: p > 0.25

with

X \sim B(20, 0.25)

, find the critical region at the

5\%

significance level.

●●●●●Level 32 marksStart

For a test of

H_0: p = 0.5

against

H_1: p < 0.5

with

X \sim B(25, 0.5)

, find the critical region at the

5\%

significance level.

●●●●●Level 32 marksStart

For a test of

H_0: p = 0.4

against

H_1: p \ne 0.4

with

X \sim B(20, 0.4)

, find the critical region at the

5\%

significance level.

●●●●●Level 33 marksStart

A test of

H_0: p = 0.25

against

H_1: p > 0.25

with

X \sim B(20, 0.25)

uses the critical region

X \ge 9

. Find the actual significance level of the test.

●●●●●Level 32 marksStart

A test of

H_0: p = 0.25

against

H_1: p > 0.25

with

X \sim B(20, 0.25)

has critical region

X \ge 9

. The observed value is

X = 10

. State, with a reason, the conclusion of the test.

●●●●●Level 31 markStart

Explain what is meant by the critical region of a hypothesis test.

●●●●●Level 31 markStart

A hotel's reception deals with requests for a replacement room key. The requests occur at random, at a mean rate of

2

per day. The manager introduces a new type of key card and believes that this will reduce the number of requests. To test this, she records the number of requests over a period of

5

days.
(a) Write down suitable null and alternative hypotheses that the manager should use.

(1 mark)
(b) Using a

5\%

level of significance, find the critical region for this test.

(2 marks)
(c) Find the actual significance level of this test.

(1 mark)

During the

5

days there are

5

requests for a replacement key.
(d) Comment on this finding in the light of your critical region.

(2 marks)

●●●●●Level 36 marksStart

The random variable

X

has a Poisson distribution with mean

\lambda

. A two-tailed test of

H_0: \lambda = 9

is to be carried out using a single observed value of

X

and a

5\%

level of significance.
(a) Find the critical region for this test. You should state the probability in each tail, which should be as close as possible to

2.5\%

(3 marks)
(b) Find the actual significance level of this test.

(2 marks)

●●●●●Level 35 marksStart

The proportion of the members of a large gym who take part in a fitness class at least once a week is

40\%

Dionne believes that the proportion of the members aged over

60

who take part in a fitness class at least once a week is different from

40\%

She takes a random sample of

24

members aged over

60

to investigate her belief.
(a) Find the critical region for a suitable test to assess Dionne's belief.

You should

state your hypotheses clearly
use a $5\%$ level of significance
state the probability of rejection in each tail

(4 marks)
(b) Write down the actual significance level of the test in part (a).

(1 mark)

In Dionne's sample,

16

of the members aged over

60

took part in a fitness class at least once a week.
(c) With reference to your answer to part (a), comment on Dionne's belief.

(1 mark)

●●●●●Level 46 marksStart

In a large town,

30\%

40

households from the town and records the number,

X

, that put out garden waste for recycling.
(a) Write down

a null hypothesis corresponding to no change in the proportion of households that put out garden waste for recycling
an alternative hypothesis corresponding to Yusuf's belief

(1 mark)
(b) Using a

5\%

level of significance, find the critical region for a two-tailed test of the null hypothesis in part (a).

You should state the probability of rejection in each tail, which should be less than

0.025

(3 marks)
(c) Find the actual level of significance of the test based on your critical region from part (b).

(1 mark)

Yusuf observes that

17

of the

40

households put out garden waste for recycling.
(d) With reference to part (b), comment on Yusuf's belief.

(1 mark)

●●●●●Level 46 marksStart

A ferry company claims that no more than 8% of its sailings leave port late. A travel journalist believes that the actual proportion is greater than 8%. In order to test the company's claim, the journalist records, for a random sample of 150 sailings, whether each one leaves port late, and carries out a hypothesis test, at the 5% significance level, using the null hypothesis

p = 0.08

.
(a) Given that the journalist's null hypothesis is correct, determine the probability that the journalist will conclude that the company's claim is incorrect.

(5 marks)
(b) The journalist finds that 17 of the 150 sailings leave port late. A colleague says

“17 out of 150 is more than 11%, so there is evidence that the actual proportion of sailings that leave port late is greater than 8%.”

Criticise this statement.

(1 mark)

●●●●●Level 46 marksStart

X

X

●●●●●Level 46 marksStart

Last year, 35% of the customers at a café paid in cash. The owner plans to test whether the proportion of customers at the café who pay in cash this year is less than 0.35. The hypotheses for the test will be

\mathrm{H}_0: p = 0.35

and

\mathrm{H}_1: p < 0.35

.

The owner chooses a random sample of 300 customers this year and notes the number

X

who pay in cash.

The random variable

Y

has a normal distribution which is a good approximation to the distribution of

X

when the null hypothesis is correct.

The significance level of the test is 2.5%.
(a) Find the value of

y

such that

\mathrm{P}(Y < y) = 0.025

.

(2 marks)
(b) Use your answer to part (a) to determine the largest value of

X

that would result in rejecting the null hypothesis.

(3 marks)

●●●●●Level 45 marksStart

In a football league, the number of goals scored in a match is modelled by a Poisson distribution with mean

2.7

. The numbers of goals scored in different matches are independent. Each round of the league consists of

10

matches.
(a) Find the probability that more than

4

goals are scored in a randomly selected match.

(2 marks)

The league introduces a new rule to encourage attacking play. The total number of goals,

T

, scored in the next round of

10

5\%

level of significance, is

T \geqslant 37

(5 marks)
(d) Using the same approximation, find the actual significance level of this test.

(2 marks)

In the next round,

35

goals are scored.
(e) Comment on this result in the light of your critical region.

(2 marks)

●●●●●Level 412 marksStart

Seeds of a certain plant germinate with probability

0.3

. A gardener plants

20

seeds in a new compost, and claims the compost increases the germination rate. Let

X

be the number of seeds that germinate, and assume

X \sim B(20, p)

.
(a) State two conditions needed for a binomial model to be suitable here. (2 marks)
(b) Assuming

p = 0.3

, find (i)

P(X = 4)

and (ii)

P(X \ge 7)

. (4 marks)
(c) Using a

5\%

level of significance, find the critical region for a test of the gardener's claim, and state the actual significance level of the test. (5 marks)
(d) In fact

11

of the

20

seeds germinate. State the conclusion of the test in context. (2 marks)

●●●●●Level 413 marksStart

A manufacturer claims that

40\%

of the customers who receive a free sample of its new product go on to buy the product. A marketing analyst believes that the true proportion is lower than this.

The analyst takes a random sample of

25

customers who received a free sample, and records the number,

X

, who went on to buy the product.
(a) Using a

5\%

level of significance, find the critical region for a test of the analyst's belief. State your hypotheses clearly, and state the actual significance level of the test. (5 marks)
(b) In the sample,

6

of the

25

customers bought the product. State the conclusion of the test in context. (2 marks)

The analyst then takes a second random sample of

300

customers who received a free sample, and finds that

104

of them bought the product.
(c) Assuming that the manufacturer's claim is true, use a suitable approximation to estimate the probability that

104

or fewer customers in a random sample of

300

buy the product. (4 marks)
(d) Comment on what your answer to part (c) suggests about the manufacturer's claim, and explain why the approximation used in part (c) is valid. (2 marks)

●●●●●Level 413 marksStart

A survey carried out in a town in 2020 found that

30\%

of adults regularly cycled to work. Since then a network of cycle lanes has opened in the town. A council officer believes that the proportion of adults in the town who regularly cycle to work has changed. To investigate, she asks

25

adults leaving the town's leisure centre one Saturday morning whether they regularly cycle to work, and uses

X

, the number who say yes, to carry out a hypothesis test.
(a) Explain why the officer should use a two-tailed test. (1 mark)
(b) Stating your hypotheses clearly, determine the critical region for the test at the

5\%

significance level, showing the probabilities you use. Find the actual significance level of the test. (5 marks)
(c) In fact

12

of the

25

adults say that they regularly cycle to work. State, with a reason, the conclusion of the test in context. (2 marks)
(d) Explain why the way the officer chose her sample makes the conclusion of the test unreliable. (1 mark)

●●●●●Level 49 marksStart

The time taken by a courier to complete a delivery round in a city,

T

minutes, is modelled by a normal distribution with mean

\mu

and standard deviation

\sigma

. Records show that

\mathrm{P}(T < 40) = 0.0668

and

\mathrm{P}(T > 58) = 0.1587

.
(a) Determine the values of

\mu

and

\sigma

. (4 marks)

The courier company introduces a route-planning app and claims that it reduces the time taken to complete a round. After the app is introduced, the times taken for a random sample of

30

rounds are recorded. The standard deviation of the times is assumed to be unchanged.
(b) Stating your hypotheses clearly, find the critical region for a test of the company's claim at the

5\%

significance level, giving your answer in terms of the sample mean

\bar{T}

. (4 marks)
(c) The mean time for the sample of

30

rounds is

48.2

minutes. Complete the test, stating your conclusion in context.

The company's analyst says that this result proves the app reduces the time taken to complete a round. Comment on the analyst's claim. (3 marks)

●●●●●Level 411 marksStart

A bus company operates the bus services in a town.

Two years ago,

35\%

of the company's passengers paid for their journey with cash.

The company has since installed contactless card readers on all of its buses.

One month after the card readers were installed,

40

passengers were selected at random and

8

of them were found to have paid for their journey with cash.

Using a binomial distribution, carry out a hypothesis test at the

5\%

significance level to investigate the claim that, since the card readers were installed, there has been a decrease in the proportion of passengers who pay for their journey with cash.

(6 marks)

●●●●●Level 46 marksStart

It is known that

25\%

of all visitors to a museum buy something from the museum's gift shop.

The museum moves its gift shop to the entrance hall.

Priya decides to investigate whether the proportion of visitors to the museum who buy something from the gift shop has changed.

Priya will carry out a hypothesis test at the

10\%

significance level on a random sample of

30

visitors to the museum.
(a) State suitable null and alternative hypotheses for Priya's test.

(1 mark)
(b) Using a

10\%

level of significance, find the critical region for Priya's test.

(5 marks)
(c) In her random sample, Priya finds

12

visitors who buy something from the gift shop.

State Priya's conclusion in context.

(1 mark)

●●●●●Level 47 marksStart

A council claims that

60\%

of the households in a town put out a recycling bin each week. A campaigner believes that the true proportion is lower than this.

The campaigner observes a random sample of

20

households in the town and records the number,

X

, that put out a recycling bin in a particular week.
(a) Using a

5\%

level of significance, find the critical region for a test of the campaigner's belief. State your hypotheses clearly. (3 marks)
(b) State the actual significance level of the test, and explain why it is not exactly

5\%

. (2 marks)
(c) In the sample,

8

households put out a recycling bin. State the conclusion of the test in context. (2 marks)

The campaigner then observes a second random sample of

150

households in the town, of which

77

put out a recycling bin.
(d) Assuming that the council's claim is correct, use a suitable approximation to estimate the probability that

77

or fewer households in a random sample of

150

put out a recycling bin. Justify the use of your approximation, and comment on what your answer suggests about the council's claim. (5 marks)

●●●●●Level 512 marksStart

A machine dispenses a liquid into bottles. The volume dispensed is normally distributed with mean

500

ml and standard deviation

4

ml.

After the machine is serviced, the manager suspects that the mean volume may have changed. She takes a random sample of

15

bottles and finds that the sample mean volume is

502.4

ml.

You may assume that the standard deviation is unchanged.
(a) Stating your hypotheses clearly, test at the

5\%

level of significance whether the mean volume dispensed has changed. (6 marks)
(b) Find the critical region for this test, giving the boundaries to

2

decimal places. (3 marks)
(c) Explain what is meant by the

5\%

level of significance in this context. (1 mark)
(d) The machine is serviced again. A further random sample of

15

bottles has a mean volume of

501.8

ml.

Explain, without further calculation, whether this sample would lead to the same conclusion. (2 marks)

●●●●●Level 512 marksStart

A machine dispenses a liquid into bottles. The volume dispensed is normally distributed with mean

500

ml and standard deviation

4

ml.

After the machine is serviced, the manager suspects that the mean volume may have changed. She takes a random sample of

15

bottles and finds that the sample mean volume is

502.4

ml.

You may assume that the standard deviation is unchanged.
(a) Stating your hypotheses clearly, test at the

5\%

level of significance whether the mean volume dispensed has changed. (6 marks)
(b) Find the critical region for this test, giving the boundaries to

2

decimal places. (3 marks)
(c) The machine is serviced again. A further random sample of

15

bottles has a mean volume of

501.8

ml.

Explain, without further calculation, whether this sample would lead to the same conclusion. (2 marks)

●●●●●Level 511 marksStart

A manufacturer claims that a treatment is effective for

60\%

of patients. In a trial, the treatment is given to a random sample of

20

patients, and

16

of them improve.

A researcher wishes to test, at the

5\%

level of significance, whether the proportion of patients for whom the treatment is effective is greater than

0.6

.
(a) Carry out the test, stating your hypotheses clearly. (5 marks)
(b) Find the critical region for this test. (4 marks)

●●●●●Level 59 marksStart

The time,

T

minutes, that a pharmacy takes to prepare a prescription is modelled by a continuous uniform distribution over the interval

[a, b]

Given that

$\mathrm{E}(T) = 16$
$\mathrm{P}(T < 11) = 0.25$

(a) find the value of

a

and the value of

b

(3 marks)
(b) Show that the probability that a prescription takes more than

22

minutes to prepare is

0.2

(1 mark)
(c) Find the exact value of

\mathrm{E}(T^2)

(2 marks)

The pharmacy introduces a new system. The manager believes that, with the new system, the proportion of prescriptions that take more than

22

minutes to prepare is less than

0.2

A random sample of

40

prescriptions prepared with the new system is to be taken.
(d) Using a

5\%

level of significance, find the critical region for a test of the manager's belief. You should state your hypotheses clearly.

(4 marks)
(e) Find the actual significance level of this test.

(1 mark)

In the sample,

3

of the prescriptions took more than

22

minutes to prepare.
(f) Comment on the manager's belief in the light of this sample. Give a reason for your answer.

(2 marks)

●●●●●Level 513 marksStart

One-tailed tests (the probability method)

38 questions

LessonNot started

A web server records errors at random, at a mean rate of

11

per day. After a software update, the server records

94

errors in the next

7

days.

Using a suitable approximation, test, at the

5\%

level of significance, whether or not there is evidence that the mean rate of errors has increased. State your hypotheses clearly.

(7 marks)

●●●●●Level 37 marksStart

A test of

H_0: p = 0.3

against

H_1: p > 0.3

uses

X \sim B(20, 0.3)

and observes

X = 10

. Using a

5\%

significance level, carry out the test.

●●●●●Level 34 marksStart

A lifeboat station is called out at random, at a mean rate of

0.7

times per week during the summer. A new coastal path opens nearby, and the crew believe that this has increased the rate of call-outs. In the first

5

weeks after the path opens, the station is called out

8

times.

Test, at the

5\%

level of significance, whether or not there is evidence to support the crew's belief. State your hypotheses clearly.

(5 marks)

●●●●●Level 35 marksStart

The time taken by a commuter train to travel between two city stations is modelled by a normal distribution with mean of

42.0

minutes and a standard deviation of

2.5

minutes.

The rail operator introduces a new timetable and claims that the mean journey time has decreased.

A random sample of

12

journeys made after the new timetable was introduced was found to have a mean journey time of

40.9

minutes.

Carry out a hypothesis test at the

5\%

significance level to investigate the rail operator's claim.

(6 marks)

●●●●●Level 36 marksStart

A geography student believes that, for towns in a hilly region, there is a positive correlation between the height of a town above sea level and the town's mean annual rainfall.

The student collected data from a random sample of

15

towns in the region.

The correlation coefficient between height above sea level and mean annual rainfall was

0.462

Carry out a hypothesis test at the

5\%

significance level to determine if this sample provides evidence to support the student's belief.

For

n = 15

, the critical value for a one-tailed test at the

5\%

level for the population correlation coefficient is

0.4409

(4 marks)

●●●●●Level 34 marksStart

At a market stall, customers are offered a taste of a cheese before they decide whether to buy any.

Over a long period, Lena has found that

40\%

of the customers who taste this cheese then buy some.

Lena changes the way the cheese is displayed on her stall and claims that the proportion of customers who taste the cheese and then buy some has increased.

A random sample of

30

customers who taste the cheese after the change is taken and

18

of them buy some cheese.

Use a suitable test to assess Lena's claim.

You should

state your hypotheses clearly
use a $5\%$ level of significance

(4 marks)

●●●●●Level 34 marksStart

p = 0.08

●●●●●Level 46 marksStart

X

X

●●●●●Level 46 marksStart

\mathrm{H}_0: p = 0.35

and

\mathrm{H}_1: p < 0.35

.

The owner chooses a random sample of 300 customers this year and notes the number

X

who pay in cash.

The random variable

Y

has a normal distribution which is a good approximation to the distribution of

X

when the null hypothesis is correct.

The significance level of the test is 2.5%.
(a) Find the value of

y

such that

\mathrm{P}(Y < y) = 0.025

.

(2 marks)
(b) Use your answer to part (a) to determine the largest value of

X

that would result in rejecting the null hypothesis.

(3 marks)

●●●●●Level 45 marksStart

●●●●●Level 47 marksStart

Bursts in the water mains of a town occur at random, at a mean rate of

7

per week.
(a) State one assumption, in context, that is needed for the number of bursts in a week to be modelled by a Poisson distribution.

(1 mark)
(b) Show that the probability of more than

10

bursts in a randomly selected week is

0.0985

4

decimal places.

(2 marks)
(c) Find the probability that each of the next

4

weeks has no more than

10

bursts.

(2 marks)

The water company then replaces the oldest pipes in the town. In the

12

weeks after this work is completed, there are

68

bursts.
(d) Using a suitable approximation, test, at the

5\%

level of significance, whether or not there is evidence that the mean rate of bursts has decreased. State your hypotheses clearly.

(7 marks)

●●●●●Level 412 marksStart

In the past the times that callers to a bank's telephone helpline spent waiting to be answered were normally distributed with mean 6.40 minutes. Last month the bank replaced the menu that callers hear before they are connected.

Following this change, the bank wishes to test whether the mean waiting time has increased. A random sample of 80 calls is chosen, and it is found that their mean waiting time is 6.83 minutes. It is given that the standard deviation of the waiting times is 1.75 minutes.

Carry out the test at the 1% significance level.

(7 marks)

●●●●●Level 47 marksStart

The length, in millimetres, of the shell of a species of land snail in the UK has population mean 24.6 and standard deviation 3.1.

A naturalist believes that the snails of this species in a particular area of woodland are larger, on average, than those in other areas of the UK.

A random sample of 40 snails of this species, taken from the woodland, has sample mean 25.5 mm.

Stating a necessary assumption, test at the 5% significance level whether the naturalist is correct.

(8 marks)

●●●●●Level 48 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 machine produces components, of which

8\%

are faulty. A random sample of

25

components is taken, and

X

is the number of faulty components in the sample.
(a) State two conditions needed for

X

to be modelled by a binomial distribution. (2 marks)
(b) Find

\mathrm{P}(X \leqslant 3)

, giving your answer to

4

decimal places. (2 marks)
(c) Find

\mathrm{P}(X \geqslant 2)

, giving your answer to

4

decimal places. (2 marks)

The machine is serviced. A new random sample of

25

components is taken, and none of them is faulty.
(d) Test, at the

5\%

level of significance, whether the proportion of faulty components has decreased. (4 marks)

●●●●●Level 410 marksStart

Seeds of a certain plant germinate with probability

0.3

. A gardener plants

20

seeds in a new compost, and claims the compost increases the germination rate. Let

X

be the number of seeds that germinate, and assume

X \sim B(20, p)

.
(a) State two conditions needed for a binomial model to be suitable here. (2 marks)
(b) Assuming

p = 0.3

, find (i)

P(X = 4)

and (ii)

P(X \ge 7)

. (4 marks)
(c) Using a

5\%

level of significance, find the critical region for a test of the gardener's claim, and state the actual significance level of the test. (5 marks)
(d) In fact

11

of the

20

seeds germinate. State the conclusion of the test in context. (2 marks)

●●●●●Level 413 marksStart

A manufacturer claims that

40\%

of the customers who receive a free sample of its new product go on to buy the product. A marketing analyst believes that the true proportion is lower than this.

The analyst takes a random sample of

25

customers who received a free sample, and records the number,

X

, who went on to buy the product.
(a) Using a

5\%

6

of the

25

customers bought the product. State the conclusion of the test in context. (2 marks)

The analyst then takes a second random sample of

300

customers who received a free sample, and finds that

104

of them bought the product.
(c) Assuming that the manufacturer's claim is true, use a suitable approximation to estimate the probability that

104

or fewer customers in a random sample of

300

buy the product. (4 marks)
(d) Comment on what your answer to part (c) suggests about the manufacturer's claim, and explain why the approximation used in part (c) is valid. (2 marks)

●●●●●Level 413 marksStart

A seed company claims that

85\%

of its parsnip seeds germinate. A gardener plants the seeds in trays of

20

, and the number of seeds in a tray that germinate is modelled by the distribution

\mathrm{B}(20,\ 0.85)

.
(a) Find the probability that, in a randomly chosen tray, exactly

18

seeds germinate. (2 marks)
(b) Find the probability that, in a randomly chosen tray, at least

17

seeds germinate. (2 marks)

The gardener suspects that the company's claim overstates the germination rate. In the next tray planted, only

13

of the

20

seeds germinate.
(c) Test, at the

5\%

significance level, whether there is evidence to support the gardener's suspicion. State your hypotheses clearly. (5 marks)

●●●●●Level 49 marksStart

A machine fills jars with honey. The mass of honey in a jar is modelled by a normal distribution with mean

410

g and standard deviation

12

g.
(a) Find the probability that a randomly chosen jar contains more than

428

g of honey. (2 marks)
(b) Determine the mass

m

such that

99\%

of jars contain less than

m

grams, giving your answer to 3 significant figures. (3 marks)

A consumer group believes the machine has drifted and is now underfilling jars. The masses of a random sample of

40

jars have mean

405.4

g. The standard deviation is assumed to remain

12

g.
(c) Test, at the

5\%

significance level, whether there is evidence that the mean mass of honey per jar has fallen below

410

g. State your hypotheses clearly, and state whether your conclusion would change at the

1\%

level. (6 marks)

●●●●●Level 411 marksStart

The time taken by a courier to complete a delivery round in a city,

T

minutes, is modelled by a normal distribution with mean

\mu

and standard deviation

\sigma

. Records show that

\mathrm{P}(T < 40) = 0.0668

and

\mathrm{P}(T > 58) = 0.1587

.
(a) Determine the values of

\mu

and

\sigma

. (4 marks)

The courier company introduces a route-planning app and claims that it reduces the time taken to complete a round. After the app is introduced, the times taken for a random sample of

30

rounds are recorded. The standard deviation of the times is assumed to be unchanged.
(b) Stating your hypotheses clearly, find the critical region for a test of the company's claim at the

5\%

significance level, giving your answer in terms of the sample mean

\bar{T}

. (4 marks)
(c) The mean time for the sample of

30

rounds is

48.2

●●●●●Level 411 marksStart

A treatment succeeds for

40\%

of patients. In a trial of

25

patients it succeeds for

6

. Test, at the

5\%

level, whether the success rate has fallen (

X \sim B(25, 0.4)

●●●●●Level 45 marksStart

A claim states

p = 0.2

. In a sample of

30

, the value

X = 11

is observed. Test

H_0: p = 0.2

against

H_1: p > 0.2

at the

5\%

level.

●●●●●Level 44 marksStart

A fair coin is suspected of favouring tails. In

15

tosses it shows

3

heads. Test

H_0: p = 0.5

against

H_1: p < 0.5

at the

5\%

level, where

p = P(\text{head})

●●●●●Level 44 marksStart

A test of

H_0: p = 0.45

against

H_1: p < 0.45

uses

X \sim B(20, 0.45)

and observes

X = 4

. Test at the

5\%

level.

●●●●●Level 44 marksStart

A hospital's maternity unit records the number of babies born each day.
(a) Explain why a Poisson distribution is unlikely to be a suitable model for the number of babies born in a day.

(1 mark)

The number of deliveries in a day, where one delivery may be of one baby or of more than one baby, is modelled by a Poisson distribution with mean

1.6

(b) State one further assumption, in context, that is needed for this model.

(1 mark)
(c) Find the probability that, on a randomly chosen day, there are

(i) no deliveries,

(ii) more than

3

deliveries.

(3 marks)

A nearby maternity unit closes, and the midwives believe that the mean number of deliveries per day has increased. In a randomly chosen period of

5

days there are

14

deliveries.
(d) Test, at the

5\%

level of significance, whether or not there is evidence to support the midwives' belief. State your hypotheses clearly.

(5 marks)

●●●●●Level 410 marksStart

Faults occur on an overhead power line at random, at a mean rate of

0.45

per month.
(a) Find the probability that there are at least

2

faults on the line in a

12

-month period.

(3 marks)

New insulators are fitted to the line. The power company claims that this has reduced the rate at which faults occur. In the

4

months after the insulators are fitted, there are no faults on the line.
(b) Test, at the

5\%

level of significance, whether or not there is evidence to support the company's claim. State your hypotheses clearly.

(5 marks)
(c) Give a reason why the test in part (b) is not an appropriate test.

(1 mark)
(d) Find the smallest whole number of months for which no faults on the line would give evidence, at the

5\%

level of significance, to support the company's claim.

(3 marks)

●●●●●Level 412 marksStart

A bus company operates the bus services in a town.

Two years ago,

35\%

of the company's passengers paid for their journey with cash.

The company has since installed contactless card readers on all of its buses.

One month after the card readers were installed,

40

passengers were selected at random and

8

of them were found to have paid for their journey with cash.

Using a binomial distribution, carry out a hypothesis test at the

5\%

significance level to investigate the claim that, since the card readers were installed, there has been a decrease in the proportion of passengers who pay for their journey with cash.

(6 marks)

●●●●●Level 46 marksStart

A workshop makes wooden beads and

\tfrac{1}{5}

of the beads are also painted gold.

The beads are packed into bags before being sent to craft shops.

There are

14

beads in each bag.

The random variable

T

represents the number of beads that are painted gold in each bag.
(a) State a condition needed for

T

to be modelled by

\mathrm{B}\!\left(14, \tfrac{1}{5}\right)

(1 mark)

A bag is selected at random.
(b) Using

T \sim \mathrm{B}\!\left(14, \tfrac{1}{5}\right)

find

(i) the probability that the bag has exactly

3

beads that are painted gold,

(ii) the probability that the bag has fewer than

2

beads that are painted gold.

(2 marks)

Iris owns a craft shop and buys

25

bags of these beads.
(c) Find the probability that exactly

4

of these

25

bags have fewer than

2

beads that are painted gold.

(2 marks)

Iris claims that the proportion of these beads that are painted gold is less than

\tfrac{1}{5}

A random sample of

60

of these beads is taken and

6

of them are painted gold.
(d) Use a suitable test to assess Iris's claim.

You should

state your hypotheses clearly
use a $5\%$ level of significance

(4 marks)

●●●●●Level 49 marksStart

A theatre has found that, over a long period,

30\%

of the people who attend a performance buy a programme.

A random sample of

25

people who attend a performance is taken.

The random variable

X

represents the number of these people who buy a programme.
(a) Using a suitable distribution for

X

, find

(i)

\mathrm{P}(X = 9)

(ii)

\mathrm{P}(X \leqslant 5)

(3 marks)

The price of a programme is then increased.

The manager of the theatre claims that the proportion of people attending a performance who buy a programme has fallen.

After the increase, a random sample of

50

people who attend a performance is taken and

9

of them buy a programme.
(b) Use a suitable test to assess the manager's claim.

You should

state your hypotheses clearly
use a $5\%$ level of significance

(4 marks)

●●●●●Level 47 marksStart

A wildlife hospital admits injured hedgehogs. During the summer, hedgehogs are admitted at random, at a mean rate of

3.5

30

admissions in a

10

-day period is

0.18

, correct to

2

decimal places.

Show your working clearly.

(Solutions relying on calculator technology are not acceptable.)

(5 marks)

In one

10

-day period there are fewer than

30

admissions.
(c) Using the same approximation and your answer to part (b), find the probability that there are fewer than

25

admissions in this period.

(3 marks)

During a heatwave,

49

hedgehogs are admitted in a

10

-day period.
(d) Using a suitable approximation, test, at the

1\%

level of significance, whether or not there is evidence that the mean rate of admissions has increased. State your hypotheses clearly.

(7 marks)

●●●●●Level 516 marksStart

Potholes form on the roads of a city at random, at a mean rate of

0.8

per km of road each winter.
(a) Find the probability that more than

2

potholes form on a

1.5

km stretch of road in one winter.

(2 marks)

The council selects a length of road,

x

km long, for a survey. When a normal approximation is used, the probability that at least

88

potholes form on this length of road in one winter is

0.8944

(b) Show that

x = 125

(7 marks)

The council resurfaces these

125

km of road. In the following winter,

82

potholes form on them.
(c) Using a suitable approximation, test, at the

5\%

level of significance, whether or not there is evidence that the mean rate at which potholes form on these roads has decreased. State your hypotheses clearly.

(7 marks)

●●●●●Level 516 marksStart

A factory makes candles of a particular type. Each day the candles are made in 11 batches, numbered 2 to 12. Ravi, the quality manager, chooses one batch each day for checking. He throws two fair six-sided dice and chooses the batch whose number is the total of the two scores.
(a) Show that this method of choosing a batch is not random.

(2 marks)

To check the burning time of a candle, the candle is lit and left to burn until it goes out.
(b) State one reason why Ravi tests a sample of candles rather than every candle that is made.

(1 mark)

In the past the burning times of candles of this type had mean 32.5 hours. The factory has recently changed to a cheaper wax, and Ravi wishes to test whether the mean burning time has decreased. He now uses a correct method to choose a random sample of 45 candles made with the cheaper wax, and finds that their mean burning time is 31.9 hours. It is given that the standard deviation of the burning times is 2.2 hours.
(c) Carry out the test at the 1% significance level.

(7 marks)

●●●●●Level 510 marksStart

A council claims that

60\%

of the households in a town put out a recycling bin each week. A campaigner believes that the true proportion is lower than this.

The campaigner observes a random sample of

20

households in the town and records the number,

X

, that put out a recycling bin in a particular week.
(a) Using a

5\%

5\%

. (2 marks)
(c) In the sample,

8

households put out a recycling bin. State the conclusion of the test in context. (2 marks)

The campaigner then observes a second random sample of

150

households in the town, of which

77

put out a recycling bin.
(d) Assuming that the council's claim is correct, use a suitable approximation to estimate the probability that

77

or fewer households in a random sample of

150

put out a recycling bin. Justify the use of your approximation, and comment on what your answer suggests about the council's claim. (5 marks)

●●●●●Level 512 marksStart

The mass,

X

grams, of a loaf of bread produced by a bakery is modelled by a normal distribution with mean

\mu

and standard deviation

\sigma

. Records show that

2.28\%

of loaves have mass less than

775

g and that

91.04\%

of loaves have mass between

775

g and

831

g.
(a) Find the values of

\mu

and

\sigma

. (5 marks)

The bakery installs a new oven. The baker believes that loaves baked in the new oven have a greater mean mass. It may be assumed that the standard deviation is unchanged. A random sample of

25

loaves baked in the new oven has mean mass

812.5

g.
(b) Test the baker's belief at the

5\%

level of significance. State your hypotheses clearly. (5 marks)
(c) Explain how, if at all, your conclusion would change if the baker had believed only that the mean mass had changed, with no view as to the direction of the change. (2 marks)
(d) A second random sample of

n

loaves from the new oven also has mean mass

812.5

g. Find the smallest value of

n

for which this sample would provide significant evidence, at the

5\%

level, that the mean mass has changed. (2 marks)

●●●●●Level 514 marksStart

A manufacturer claims that a treatment is effective for

60\%

of patients. In a trial, the treatment is given to a random sample of

20

patients, and

16

of them improve.

A researcher wishes to test, at the

5\%

level of significance, whether the proportion of patients for whom the treatment is effective is greater than

0.6

.
(a) Carry out the test, stating your hypotheses clearly. (5 marks)
(b) Find the critical region for this test. (4 marks)

●●●●●Level 59 marksStart

A city bike-hire scheme charges

£2

for a short ride,

£4

for a medium ride and

£7

for a long ride. The operator models the rides taken as short, medium and long in the ratio

2 : 1 : 1

respectively.

A random sample of

2

rides is taken from the very large number of rides taken. The random variable

R

represents the range, in pounds, of the charges for the

2

rides.
(a) Show that, according to the model,

P(R = 5) = \frac14

(2 marks)
(b) Find the sampling distribution of

R

(4 marks)
(c) Find

E(R)

(2 marks)

The operator suspects that samples of

2

rides with a range of

£5

occur more often than the model suggests. She takes

20

random samples of

2

rides and finds that

9

of these samples have a range of

£5

(d) Test, at the

5\%

level of significance, whether or not there is evidence to support the operator's suspicion. State your hypotheses clearly.

(5 marks)

●●●●●Level 513 marksStart

Historically

15\%

of seeds fail to germinate. A new batch of

40

seeds has

11

failures. Test, at the

5\%

level, whether the failure rate has increased (

X \sim B(40, 0.15)

●●●●●Level 55 marksStart

The time,

T

minutes, that a pharmacy takes to prepare a prescription is modelled by a continuous uniform distribution over the interval

[a, b]

Given that

$\mathrm{E}(T) = 16$
$\mathrm{P}(T < 11) = 0.25$

(a) find the value of

a

and the value of

b

(3 marks)
(b) Show that the probability that a prescription takes more than

22

minutes to prepare is

0.2

(1 mark)
(c) Find the exact value of

\mathrm{E}(T^2)

(2 marks)

The pharmacy introduces a new system. The manager believes that, with the new system, the proportion of prescriptions that take more than

22

minutes to prepare is less than

0.2

A random sample of

40

prescriptions prepared with the new system is to be taken.
(d) Using a

5\%

3

of the prescriptions took more than

22

minutes to prepare.
(f) Comment on the manager's belief in the light of this sample. Give a reason for your answer.

(2 marks)

●●●●●Level 513 marksStart

Two-tailed tests

25 questions

LessonNot started

In a large city,

45\%

of journeys to work are made by car.

Priya believes that the proportion of journeys to work made by car by people who live in the city centre is different from

45\%

She takes a random sample of

20

people who live in the city centre and records the number,

X

, who travel to work by car.
(a) Write down

a null hypothesis corresponding to no change in the proportion
an alternative hypothesis corresponding to Priya's belief

(1 mark)
(b) Using a

5\%

level of significance, find the critical region for a two-tailed test of the null hypothesis in part (a).

You should state the probability of rejection in each tail, which should be less than

0.025

(3 marks)

●●●●●Level 34 marksStart

A bird-ringing station catches a rare species of warbler at random, at a mean rate of

1.9

per day during October. This October,

4

of these warblers were caught in the first

5

days of the month.

Test, at the

5\%

level of significance, whether or not the mean rate at which these warblers are caught has changed. State your hypotheses clearly.

(5 marks)

●●●●●Level 35 marksStart

The random variable

X

has a Poisson distribution with mean

\lambda

. A two-tailed test of

H_0: \lambda = 9

is to be carried out using a single observed value of

X

and a

5\%

level of significance.
(a) Find the critical region for this test. You should state the probability in each tail, which should be as close as possible to

2.5\%

(3 marks)
(b) Find the actual significance level of this test.

(2 marks)

●●●●●Level 35 marksStart

A coin is tossed

20

times and shows

15

heads. Test, at the

5\%

level, whether the coin is biased (

H_0: p = 0.5

H_1: p \ne 0.5

●●●●●Level 34 marksStart

A claim states

p = 0.3

. In a sample of

25

X = 3

is observed. Test

H_0: p = 0.3

against

H_1: p \ne 0.3

at the

5\%

level.

●●●●●Level 34 marksStart

A coin is tossed

40

times and shows

26

heads. Test, at the

5\%

level, whether the coin is biased.

●●●●●Level 34 marksStart

A test of

H_0: p = 0.4

against

H_1: p \ne 0.4

uses

X \sim B(30, 0.4)

and observes

X = 6

. Test at the

5\%

level.

●●●●●Level 34 marksStart

Explain how a two-tailed test at the

5\%

significance level is carried out differently from a one-tailed test at the same level.

●●●●●Level 32 marksStart

A coin is tossed

20

times and shows

5

heads. Test, at the

5\%

level, whether the coin is biased.

●●●●●Level 34 marksStart

The proportion of the members of a large gym who take part in a fitness class at least once a week is

40\%

Dionne believes that the proportion of the members aged over

60

who take part in a fitness class at least once a week is different from

40\%

She takes a random sample of

24

members aged over

60

to investigate her belief.
(a) Find the critical region for a suitable test to assess Dionne's belief.

You should

state your hypotheses clearly
use a $5\%$ level of significance
state the probability of rejection in each tail

(4 marks)
(b) Write down the actual significance level of the test in part (a).

(1 mark)

In Dionne's sample,

16

of the members aged over

60

took part in a fitness class at least once a week.
(c) With reference to your answer to part (a), comment on Dionne's belief.

(1 mark)

●●●●●Level 46 marksStart

In a large town,

30\%

40

households from the town and records the number,

X

, that put out garden waste for recycling.
(a) Write down

a null hypothesis corresponding to no change in the proportion of households that put out garden waste for recycling
an alternative hypothesis corresponding to Yusuf's belief

(1 mark)
(b) Using a

5\%

level of significance, find the critical region for a two-tailed test of the null hypothesis in part (a).

You should state the probability of rejection in each tail, which should be less than

0.025

(3 marks)
(c) Find the actual level of significance of the test based on your critical region from part (b).

(1 mark)

Yusuf observes that

17

of the

40

households put out garden waste for recycling.
(d) With reference to part (b), comment on Yusuf's belief.

(1 mark)

●●●●●Level 46 marksStart

The thickness, in millimetres, of a sheet of glass made by a particular machine is denoted by the random variable

X

, where

X

has the distribution

\mathrm{N}(\mu, 0.0144)

.

In the past the value of

\mu

has been

6.00

.

After the machine was serviced, the mean thickness of

36

randomly chosen sheets of glass made by the machine was found to be

6.04

millimetres.

Test, at the 5% significance level, whether the mean thickness of the sheets of glass has changed.

(7 marks)

●●●●●Level 47 marksStart

A drinks machine dispenses cups of coffee. The volume dispensed, in ml, is modelled by a normal distribution with standard deviation

15

. For a long period the mean volume has been

330

ml.

After the machine is serviced, the operator suspects the mean volume has changed. A random sample of

25

cups has mean volume

336.6

ml.
(a) Write down the distribution of the sample mean

\bar{X}

under the assumption that the mean is still

330

ml. (2 marks)
(b) Test, at the

5\%

significance level, whether the mean volume has changed. State your hypotheses clearly, and give the

p

-value of your sample. (7 marks)
(c) The operator's colleague believes the service could only have INCREASED the mean volume. State the critical value of

z

for the corresponding one-tailed test at the

5\%

level, the conclusion of that test, and its

p

-value. (3 marks)
(d) State one assumption about the population standard deviation used in these tests, and interpret what "the 5% significance level" means for the probability of an error. (2 marks)

●●●●●Level 414 marksStart

A survey carried out in a town in 2020 found that

30\%

25

adults leaving the town's leisure centre one Saturday morning whether they regularly cycle to work, and uses

X

5\%

significance level, showing the probabilities you use. Find the actual significance level of the test. (5 marks)
(c) In fact

12

of the

25

●●●●●Level 49 marksStart

The mass,

M

grams, of a loaf of bread produced by a bakery is modelled by a normal distribution with mean

\mu

and standard deviation

\sigma

. Records show that

10.56\%

of loaves have a mass less than

780

g and

3.04\%

have a mass greater than

830

g.
(a) Find the values of

\mu

and

\sigma

. (5 marks)

The bakery installs a new oven. A random sample of

40

loaves baked in the new oven has a mean mass of

795.2

g. It is assumed that the standard deviation of the masses is unchanged.
(b) Test, at the

5\%

level of significance, whether the mean mass of loaves baked in the new oven is different from the value of

\mu

found in part (a). (4 marks)
(c) The manager says, \"The test proves that the new oven has not changed the mean mass.\" Explain why this statement is not justified. (1 mark)

●●●●●Level 410 marksStart

Light aircraft land at a small airfield at random, at a mean rate of

1.1

per hour between 10 am and 3 pm on weekdays.
(a) State two assumptions, in context, that are needed for a Poisson distribution to model the number of landings between 10 am and 3 pm on a weekday.

(2 marks)

The airfield lowers its landing fees. The airfield's operator wants to know whether the mean rate of landings between 10 am and 3 pm on weekdays has changed. One weekday is chosen at random, and the number of landings between 10 am and 3 pm is recorded.
(b) Write down suitable null and alternative hypotheses for the operator's test.

(1 mark)
(c) Using a

3\%

level of significance, find the critical region for this test.

(3 marks)
(d) Find the actual significance level of this test.

(1 mark)

On the chosen day there are

10

landings between 10 am and 3 pm.
(e) Comment on this finding in the light of your critical region.

(2 marks)

The operator also records the number of passengers who arrive at the airfield on these aircraft.
(f) Explain why a Poisson distribution is unlikely to be a suitable model for the number of passengers arriving between 10 am and 3 pm.

(1 mark)

●●●●●Level 410 marksStart

Knots occur at random along lengths of timber from a particular supplier, at a mean rate of

1.5

per metre.
(a) Find the probability that a

2

-metre plank from this supplier has no knots.

(2 marks)

A carpenter changes to a new supplier. He wants to test whether the mean rate at which knots occur is different for the new supplier's timber. He will count the knots in a randomly chosen

5

-metre length of timber from the new supplier.
(b) Write down suitable null and alternative hypotheses for the carpenter's test.

(1 mark)
(c) Using a

5\%

level of significance, find the critical region for this test. You should state the probability in each tail, which should be as close as possible to

2.5\%

(3 marks)
(d) Write down the actual significance level of this test.

(1 mark)

The

5

-metre length of timber from the new supplier has

14

knots.
(e) Comment on this finding in the light of your critical region.

(2 marks)

●●●●●Level 49 marksStart

It is known that

25\%

10\%

significance level on a random sample of

30

visitors to the museum.
(a) State suitable null and alternative hypotheses for Priya's test.

(1 mark)
(b) Using a

10\%

level of significance, find the critical region for Priya's test.

(5 marks)
(c) In her random sample, Priya finds

12

visitors who buy something from the gift shop.

State Priya's conclusion in context.

(1 mark)

●●●●●Level 47 marksStart

During weekday evenings, requests for rides arrive at a taxi company's app at random, at a mean rate of

r

per minute.

When a normal approximation is used, the probability of at least

60

requests in a

20

-minute period is

0.0668

(a) Show that

r = 2.45

(7 marks)

Of the requests,

20\%

are for journeys to the airport, independently of each other.
(b) Find the probability that there are fewer than

2

requests for journeys to the airport in a

5

-minute period.

(3 marks)

The company then raises its evening fares. In a randomly selected hour after the rise, there are

128

requests.
(c) Using a suitable approximation, test, at the

5\%

level of significance, whether or not there is evidence that the mean rate of requests has changed. State your hypotheses clearly.

(7 marks)

●●●●●Level 517 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

The mass,

X

grams, of a loaf of bread produced by a bakery is modelled by a normal distribution with mean

\mu

and standard deviation

\sigma

. Records show that

2.28\%

of loaves have mass less than

775

g and that

91.04\%

of loaves have mass between

775

g and

831

g.
(a) Find the values of

\mu

and

\sigma

25

loaves baked in the new oven has mean mass

812.5

g.
(b) Test the baker's belief at the

5\%

n

loaves from the new oven also has mean mass

812.5

g. Find the smallest value of

n

for which this sample would provide significant evidence, at the

5\%

level, that the mean mass has changed. (2 marks)

●●●●●Level 514 marksStart

A machine dispenses a liquid into bottles. The volume dispensed is normally distributed with mean

500

ml and standard deviation

4

ml.

After the machine is serviced, the manager suspects that the mean volume may have changed. She takes a random sample of

15

bottles and finds that the sample mean volume is

502.4

ml.

You may assume that the standard deviation is unchanged.
(a) Stating your hypotheses clearly, test at the

5\%

level of significance whether the mean volume dispensed has changed. (6 marks)
(b) Find the critical region for this test, giving the boundaries to

2

decimal places. (3 marks)
(c) Explain what is meant by the

5\%

level of significance in this context. (1 mark)
(d) The machine is serviced again. A further random sample of

15

bottles has a mean volume of

501.8

ml.

Explain, without further calculation, whether this sample would lead to the same conclusion. (2 marks)

●●●●●Level 512 marksStart

A machine dispenses a liquid into bottles. The volume dispensed is normally distributed with mean

500

ml and standard deviation

4

ml.

After the machine is serviced, the manager suspects that the mean volume may have changed. She takes a random sample of

15

bottles and finds that the sample mean volume is

502.4

ml.

You may assume that the standard deviation is unchanged.
(a) Stating your hypotheses clearly, test at the

5\%

level of significance whether the mean volume dispensed has changed. (6 marks)
(b) Find the critical region for this test, giving the boundaries to

2

decimal places. (3 marks)
(c) The machine is serviced again. A further random sample of

15

bottles has a mean volume of

501.8

ml.

Explain, without further calculation, whether this sample would lead to the same conclusion. (2 marks)

●●●●●Level 511 marksStart

A mountain-rescue team is called out at random. The number of call-outs in a week is modelled by a Poisson distribution with mean

\lambda

. Records show that the probability of no call-outs in a week is

0.1827

(a) Show that

\lambda = 1.7

2

significant figures.

(2 marks)

In the rest of this question use

\lambda = 1.7

New signs are put up on the mountain paths. The team leader wants to know whether the mean rate of call-outs has changed. She will use the number of call-outs in the

5

weeks after the signs are put up.
(b) Using a

3\%

level of significance, find the critical region for this test. State your hypotheses clearly.

(4 marks)
(c) Find the actual significance level of this test.

(2 marks)

In the

5

weeks after the signs are put up, there are

3

call-outs.
(d) Comment on this finding in the light of your critical region.

(2 marks)

The team leader now believes that the signs have reduced the mean rate of call-outs. In the next

4

weeks there is

1

call-out.
(e) Test, at the

5\%

level of significance, whether or not there is evidence to support the team leader's belief. State your hypotheses clearly.

(5 marks)

●●●●●Level 515 marksStart

The mass of flour used each day by a bakery may be modelled by a normal distribution with mean

86

kg and standard deviation

7

kg.
(a) State the probability that the mass of flour used on any given day is exactly

86

kg.

(1 mark)
(b) To attract more customers, the bakery decides to change the range of bread that it bakes.

Following the decision, it was found that over a

15

-day period a total mass of

1353

kg of flour was used.

It can be assumed that the distribution of the mass of flour used each day is still normal with standard deviation

7

kg, and that the

15

-day period can be regarded as a random sample.

Investigate, at the

5\%

level of significance, whether the mean daily mass of flour used has changed.

(7 marks)
(c) The bakery's manager claims that if a different sample of

15

days had been used, the hypothesis test in part (b) would have given the same result.

Comment on the validity of this claim.

(2 marks)

●●●●●Level 510 marksStart

Hypothesis testing for zero correlation

28 questions

LessonNot started

Tomas is investigating whether there is a relationship between the mass of a pumpkin and the number of seeds it contains.

He takes a random sample of

16

pumpkins from a large field and records, for each pumpkin,

its mass, $m$ kg
the number of seeds it contains, $s$

The product moment correlation coefficient between

m

and

s

0.3915

The critical value for a sample of size

16

at the

5\%

level of significance is

0.4259

Test whether or not there is evidence of a positive correlation between the mass of a pumpkin and the number of seeds it contains.

You should

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

(3 marks)

●●●●●Level 33 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

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

A sample of

n = 10

gives a PMCC of

r = 0.72

. Test, at the

5\%

significance level, whether there is positive correlation in the population. The critical value is

0.5494

●●●●●Level 33 marksStart

A sample of

n = 12

gives

r = -0.62

. Test at the

5\%

level whether there is correlation in the population. For this two-tailed test the critical value is

0.5760

●●●●●Level 33 marksStart

A sample of

n = 10

gives

r = 0.40

. Test at the

5\%

level whether there is positive correlation. The critical value is

0.5494

●●●●●Level 33 marksStart

For

5

pairs of data,

\sum x = 15

\sum y = 20

\sum xy = 69

\sum x^2 = 55

\sum y^2 = 90

.
(a) Show that

r = 0.900

.
(b) Test at the

5\%

level for positive correlation. The critical value is

0.8054

●●●●●Level 35 marksStart

A sample of

n = 15

pairs gives a product moment correlation coefficient of

r = 0.52

.

Test, at the

5\%

significance level, whether there is evidence of positive correlation in the population. The critical value for

n = 15

0.4409

. (4 marks)

●●●●●Level 34 marksStart

A sample of

n = 25

pairs gives

r = 0.30

.

Test, at the

5\%

significance level, whether there is evidence of positive correlation. The critical value for

n = 25

0.3365

. (4 marks)

●●●●●Level 34 marksStart

A sample of

n = 8

pairs gives

r = 0.65

.

Test, at the

5\%

significance level, whether there is evidence of positive correlation. The critical value for

n = 8

0.6215

. (4 marks)

●●●●●Level 34 marksStart

A student is testing whether taller people tend to have larger hand spans, using a sample of

20

people.
(a) State suitable hypotheses for the test, defining any symbol you use. (2 marks)
(b) Explain the difference between

r

and

\rho

in this context. (1 mark)

●●●●●Level 33 marksStart

A researcher will test

H_{0}: \rho = 0

against

H_{1}: \rho > 0

at the

5\%

level using a sample of

n = 20

pairs. The critical value for

n = 20

0.3783

.

State the range of values of the sample PMCC

r

that would lead to rejecting

H_{0}

, and the name given to this set of values. (2 marks)

●●●●●Level 32 marksStart

15

towns in the region.

The correlation coefficient between height above sea level and mean annual rainfall was

0.462

Carry out a hypothesis test at the

5\%

significance level to determine if this sample provides evidence to support the student's belief.

For

n = 15

, the critical value for a one-tailed test at the

5\%

level for the population correlation coefficient is

0.4409

(4 marks)

●●●●●Level 34 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 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 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

A sample of

n = 10

gives

r = -0.55

. Test at the

5\%

level whether there is negative correlation in the population. The critical value is

0.5494

●●●●●Level 43 marksStart

For

8

pairs of data the PMCC is found to be

r = 0.956

. Test at the

5\%

level whether there is correlation in the population. For this two-tailed test with

n = 8

the critical value is

0.7067

●●●●●Level 43 marksStart

A sample of

n = 20

gives

r = 0.45

. Test at the

5\%

level whether there is positive correlation. The critical value is

0.3783

●●●●●Level 43 marksStart

A sample of

n = 12

pairs gives

r = -0.61

.

Test, at the

5\%

significance level, whether there is evidence of ANY correlation (positive or negative). The two-tail critical value for

n = 12

0.5760

. (4 marks)

●●●●●Level 44 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

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

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)