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

Critical regions and actual significance level

28 questions

LessonNot started

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

One-tailed tests (the probability method)

38 questions

LessonNot started

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

Two-tailed tests

25 questions

LessonNot started

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

Hypothesis testing for zero correlation

28 questions

LessonNot started

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

He measures the depth of the water,

y

metres, at time

t

days after 1 June, for

12

values of

t

between

t = 0

and

t = 88

Callum finds the equation of the regression line of

y

t

for his data to be

y = 8.13 - 0.0241t

(a) Interpret the gradient of this line.

(1 mark)

The product moment correlation coefficient between

y

and

t

-0.6602

The critical value for a sample of size

12

at the

5\%

level of significance is

0.4973

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

You should

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

(3 marks)

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

55

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

y = 8.13 - 0.0241t

is an appropriate model for these data.

(1 mark)

Nina suggests an improved model using the variable

u = (t - k)^2

, where

k

is a constant.

She obtains the equation

y = 6.10 + 0.00111u

(d) Choose a suitable value for

k

to write Nina's improved model for

y

in terms of

t

only.

(1 mark)

●●●●●Level 46 marksStart

A 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)