AQA A Level Maths: Statistics

Topic Questions

5.1 Hypothesis Testing

1
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5 marks

A hypothesis test uses a sample of data in an experiment to test a statement made about the value of a population parameter (p).  

Explain, in the context of hypothesis testing, what is meant by:

(i)
‘sample of data’,  

(ii)
 ‘population parameter’


(iii)    ‘null hypothesis’,

(iv)    ‘alternative hypothesis’,

(v)     ‘test statistic’.

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2a
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4 marks

From previous research, Marta has found that in general there is a 15% chance that any given customer ordering food at her restaurant will choose a salad.  She wants to test whether people are more inclined to eat salads when it is sunny out.

(i)
Clearly defining the value of the population parameter (p), state a suitable null hypothesis that Marta could use for this test.

(ii)
State a suitable alternative hypothesis that Marta could use for this test.

(iii)
Give an example of a test statistic that Marta could use to carry out this test.
2b
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1 mark

After carrying out the test, Marta had evidence to conclude that people are more likely to eat salads when the sun is out. State whether she accepted or rejected the null hypothesis you have written in part (a)(i).

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3
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6 marks

For the following null and alternative hypotheses, state whether the test is a one-tailed or a two-tailed test and give a suitable example context for each problem.

(i)
straight H subscript 0 ∶ p equals 0.5 comma space straight H subscript 1 ∶ p greater than 0.5.

(ii)
straight H subscript 0 ∶ p equals 1 over 6 comma space straight H subscript 1 ∶ p not equal to 1 over 6.

(iii)
H subscript 0 ∶ p equals 0.3 comma space H subscript 1 ∶ p less than 0.3.

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4
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6 marks

In a quiz, students have to choose the correct answer to each question from three possible options.  There is only one correct answer for each question.  Ethan got k answers correct, and he claims that he merely guessed the answer to every question but his teacher believes he used some knowledge in the quiz.  She uses the null hypothesis  straight H subscript 0 ∶ p equals 1 third  to test her belief at the 10% significance level.

(i)
If the teacher wishes to test to see if Ethan was trying to get the answers correct, rather than guessing them at random, write down the alternative hypothesis she should use and explain the conditions under which the null hypothesis would be rejected.

(ii)
If the teacher wishes to test to see if Ethan was trying to get the answers incorrect, rather than guessing them at random, write down the alternative hypothesis she should use and explain the conditions under which the null hypothesis would be rejected.

(iii)
If the teacher wishes to test to see whether Ethan was not guessing the answers at random, but she is uncertain whether he was using his knowledge to get them right or to get them wrong, write down the alternative hypothesis she should use and explain the conditions under which the null hypothesis would be rejected.

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5a
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4 marks

A hypothesis test at the 4% significance level is carried out on a spinner with four sectors using the following hypotheses:

straight H subscript 0 ∶ p equals 1 fourth comma space straight H subscript 1 ∶ p not equal to 1 fourth comma

(i)
Describe what the parameter, p, could be defined as.

(ii)
In the context of this question, explain how the significance level of 4% should be used.

(iii)
If the significance level were instead given as 10%, would the probability of incorrectly rejecting the null hypothesis be likely to increase or decrease? Give a reason for your answer.
5b
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2 marks

The spinner is spun 50 times and it is decided to reject the null hypothesis if there are less than 7 or more than 18 successes.

(i)
The critical regions for this test are given as  X less or equal than a  and  X greater or equal than b.  Write down the values of a and b.

(ii)
State the set of values for which the null hypothesis would be accepted.

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6a
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2 marks

Two volunteers at a national park, Owen and Cathy, have begun a campaign to stop people leaving their litter behind after visiting the park.  To see whether their campaign has had an effect, Owen conducts a hypothesis test at the 10% significance level, using the following hypotheses:

straight H subscript 0 ∶ p equals 0.2 comma space space space space space straight H subscript 1 ∶ p not equal to 0.2

(i)
State the percentage of people who left litter behind in the national park before the start of the campaign.

(ii)
State whether this is a one-tailed or two-tailed test.  
6b
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2 marks

Owen observes a random sample of 100 people at the national park and finds that 14 of them left litter behind.  He calculates that if  straight H subscript 0were true, then the probability of 14 or less people leaving litter would be 0.08044.

With reference to the hypotheses above, state with a reason whether Owen should accept or reject his null hypothesis.

6c
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2 marks

Cathy conducted her own hypothesis test at the 10% significance level, using the same sample data as Owen, but instead she used the following hypotheses:

 straight H subscript 0 ∶ p equals 0.2 comma space space space space straight H subscript 1 ∶ p less than 0.2

(i)
Explain how Cathy’s hypothesis test is different to Owen’s.

(ii)
Using these hypotheses, state whether the sample results given in part (b) should lead Cathy to accept or reject her null hypothesis. Give a reason for your answer.

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7a
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2 marks

A drinks manufacturer, BestBubbles, claims that in taste tests more than 50% of people can distinguish between its drinks and those of a rival brand.  The company decides to test its claim by having 20 people each taste two drinks and then attempt to determine which was made by BestBubbles and which was made by the rival company. The random variable X represents the number of people who correctly identify the drink that was made by BestBubbles.

(i)
State, giving a reason, whether this is a one-tailed or a two-tailed test.

(ii)
Write down the null and alternative hypotheses for this test.
7b
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5 marks

Under the null hypothesis, it is given that:

straight P left parenthesis X equals 13 right parenthesis equals 0.07393

straight P left parenthesis X equals 14 right parenthesis equals 0.03696

straight P left parenthesis X greater than 14 right parenthesis equals 0.02069

(i)
Calculate straight P left parenthesis X greater or equal than 14 right parenthesis and straight P left parenthesis X greater or equal than 13 right parenthesis.

(ii)
Given that a 10% level of significance was used, write down the critical value and the critical region for this test.

(iii)
State the actual level of significance for this test.       
7c
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2 marks

In fact, 15 of the 20 people correctly identify the drink made by BestBubbles.

(i)
State whether there is sufficient evidence to reject the null hypothesis at the 10% significance level.

(ii)
Write a conclusion for this hypothesis test in the context of the question.

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1a
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4 marks

Joel is a manager at a swimming pool and claims that less than half of customers wear goggles in the water.  Joel forms a sample using the next 100 swimmers and he notes that 42 of them wear goggles.

If X tilde B left parenthesis 100 comma 0.5 right parenthesis then:

straight P left parenthesis X less than 42 right parenthesis equals 0.0443

straight P left parenthesis X less or equal than 42 right parenthesis equals 0.0666

straight P left parenthesis X equals 42 right parenthesis equals 0.0223

straight P left parenthesis X greater or equal than 42 right parenthesis equals 0.9557

straight P left parenthesis X greater than 42 right parenthesis equals 0.9334

Stating your hypotheses clearly, test Joel’s claim using a 5% level of significance.

1b
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2 marks

Joel discovers that there was a family of 12 people included in the sample, all of whom wore goggles.

Explain how this information affects the conclusion to the hypothesis test.

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2a
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4 marks

At Hilbert’s Hotel three quarters of customers leave feedback upon departure by writing a comment in a book on the reception desk.  Karla, the manager, decides to get rid of the feedback book and instead leaves a feedback form in each room.  To test whether this new system has made a difference to the proportion of guests who leave feedback, Karla forms a sample using the next 80 room bookings.  Once the 80 sets of guests leave Hilbert’s Hotel, Karla counts that 65 feedback forms have been completed.

When X tilde B left parenthesis 80 comma space 3 over 4 right parenthesis space space spacethe following probabilities are given:

straight P left parenthesis X less than 65 right parenthesis equals 0.8792

straight P left parenthesis X less or equal than 65 right parenthesis equals 0.9260

straight P left parenthesis X equals 65 right parenthesis equals 0.0468

straight P left parenthesis X greater or equal than 65 right parenthesis equals 0.1208

straight P left parenthesis X greater than 65 right parenthesis equals 0.0740

Test, using a 10% level of significance, whether there is evidence to suggest that the feedback forms have changed the proportion of guests who leave feedback. State your hypotheses clearly.

2b
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2 marks

Karla repeats the same test, with the same hypotheses, the following week and finds that 53 out of the 80 sets of guests fill in their feedback forms.  This leads to the null hypothesis being rejected.  Karla claims that this shows that there is evidence that the proportion of guests leaving feedback has decreased.

Explain whether Karla’s claim is valid.

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3a
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1 mark

Explain one advantage of using critical regions instead of straight p-values for a hypothesis test.

3b
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2 marks

A test of the null hypothesis  straight H subscript 0 colon p equals 0.2 against the alternative hypothesis straight H subscript 1 colon p less than 0.2 is carried out for the random variable X tilde B left parenthesis n comma p right parenthesis

x  straight P left parenthesis X equals x right parenthesis
0 0.000406
1 0.003549
2 0.015085
3 0.041484
4 0.082968


The table below shows the probabilities for different values that space X tilde B left parenthesis n comma 0.2 right parenthesis spacecan take:

Calculate the p-value for the test statistic space x equals 3.

3c
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3 marks
(i)
Using a 5% level of significance, find the critical region for the test.

(ii)
State the actual level of significance for the test.

 

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4a
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3 marks

A group of high school statistics students are investigating the probability of winning a game called Chi Squares.  Their teacher claims that they have more than a 60% chance of winning the game.  To test the claim, they play 30 games of Chi Squares and win 80% of them.  They perform a hypothesis test using a 5% level of significance.  Below are shown the solutions of two students, Gertrude and Nate:

Gertrude’s solution

straight H subscript 0 ∶ p equals 0.6

straight H subscript 1 ∶ p greater or equal than 0.6


Nate’s solution

straight H subscript 0 ∶ p equals 0.6

straight H subscript 1 ∶ p greater than 24 over 30 equals 0.8

Let X be the number of games won,
X tilde B left parenthesis 30 comma 0.6 right parenthesis

straight P left parenthesis X equals 24 right parenthesis equals 0.0115

0.0115 less than 0.05 do not reject straight H subscript 0

Let  be the number of games won,
X tilde B left parenthesis 30 comma 0.6 right parenthesis

straight P left parenthesis X greater than 24 right parenthesis equals 0.0057

0.0057 less than 0.05 so reject straight H subscript 0


You are given that the students have correctly calculated their probabilities.

Identify and explain the three mistakes made by Gertrude.

4b
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2 marks

Identify and explain the two mistakes made by Nate.

4c
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1 mark

Use the information above to find the p-value for the test statistic space x equals 24, showing your calculation clearly.

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5a
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1 mark

Explain what you understand by the significance level of a hypothesis test.

5b
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2 marks
(i)
Give an advantage of using a lower significance level for a hypothesis test.

(ii)
Give a disadvantage of using a lower significance level for a hypothesis test.
5c
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6 marks

For each of the following scenarios, explain whether a 1%, 5% or 10% level of significance would be most appropriate.

(i)
A shopkeeper takes a sample of 10 cartons of milk to test whether the amount of milk in a carton has decreased.

(ii)
A doctor takes a sample of 100 patients to test whether there is an improvement to the recovery rate of an illness when a new drug is used, compared with the current best treatment regime.

(iii)
A manager takes a sample of 100 employees to test whether their level of job satisfaction has changed after new working hours have been introduced.

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6a
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3 marks

The table below shows the cumulative probabilities for different values that X tilde B left parenthesis 10 comma 0.5 right parenthesis space can take:

x  straight P left parenthesis X less or equal than x right parenthesis
0 0.000977
1 0.010742
2 0.054688
3 0.171875
4 0.376953
5 0.623047


Kieran collects coins and suspects that one of them is biased.  To test his suspicion Kieran flips the coin 10 times and records the number of times, T, that it lands on tails.

Stating your hypotheses clearly, find the critical regions for the test using a 10% level of significance.

6b
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2 marks

Calculate the probability of incorrectly rejecting the null hypothesis.

6c
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1 mark

The coin lands on heads on each of the 10 flips.  Kieran claims that the coin is definitely biased.

Comment on the validity of Kieran’s claim.

6d
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1 mark

Describe one adjustment Kieran could make to his test to give a more reliable conclusion.

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1a
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2 marks

Explain what you understand by a critical region of a test statistic.

1b
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2 marks

Nationally 44% of A Level mathematics students identify as female.  The headteacher of a particular school claims that the proportion of A Level mathematics students in the school who identify as female is higher than the national average.

(i)
State a suitable null hypothesis to test the headteacher’s claim.

(ii)
State a suitable alternative hypothesis to test the headteacher’s claim.
1c
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2 marks

The headteacher takes a random sample of 60 A Level mathematics students and records the number of them who identify as female, x.  For a test at the 10% significance level the critical region is X greater or equal than 32.

Given that space x equals 36, comment on the headteacher’s claim.

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2a
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2 marks

The probability of a chicken laying an egg on any given day is 65%.  Two farmers, Amina and Bert, have 30 chickens each.  They believe that the probability of their chickens laying an egg on any given day is different to 65%.

(i)
State a suitable null hypothesis to test the farmers’ belief.

(ii)
State a suitable alternative hypothesis for a two-tailed test.
2b
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1 mark

During a specific day, Amina and Bert each record the number of their 30 chickens that lay an egg.  At the 5% significance level the critical regions for this test are X less or equal than 13 and space X greater or equal than 25..

Write down the critical values for the hypothesis test.

2c
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4 marks
(i)
Given that for Amina space x equals 12, comment on her belief.

(ii)
Given that for Bert space x equals 24, comment on his belief.

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3a
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2 marks

A memory experiment involves having participants read a list of 20 words for two minutes and then recording how many of the words they can recall.  Peter, a psychologist, claims that more than 60% of teenagers can recall all the words.  Peter takes a random sample of 40 teenagers and records how many of them recall all the words.

(i)
State a suitable null hypothesis to test the psychologist’s claim.

(ii)
State a suitable alternative hypothesis to test the psychologist’s claim.
3b
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3 marks

Given that the critical value for the test is x equals 19, state the outcome of the test if

(i)
18 out of the 40 teenagers recall all the words

(ii)
19 out of the 40 teenagers recall all the words

(iii)
20 out of the 40 teenagers recall all the words.

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4a
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2 marks

A machine produces toys for a company.  It was found that 8% of the toys it was producing were faulty.  After an engineer works on the machine, she claims that the proportion of faulty toys should now have decreased.

State suitable null and alternative hypotheses to test this claim.

4b
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2 marks

After the engineer is finished, the manager of the company takes a random sample of 100 toys and finds that 2 of them are faulty.  

Given that space straight P left parenthesis X less or equal than 2 right parenthesis equals 0.01127 space when X tilde B left parenthesis 100 comma 0.08 right parenthesis, determine the outcome of the hypothesis test using a 1% level of significance. Give your conclusion in context.

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5a
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2 marks

After it was estimated that only 72% of patients were turning up for their appointments at Pearly Teeth dental surgery, the owner began sending text message reminders to the patients on the day before their appointments.  In order to test whether the reminders have increased the proportion of patients turning up to their appointments, the owner decides to conduct a hypothesis test at the 5% level of significance using the next 160 patients scheduled for appointments as a sample.

State suitable null and alternative hypotheses to test this claim.

5b
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1 mark

Describe, in context, the test statistic for this test.

5c
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2 marks

Out of the 160 patients used for the sample, 127 turned up for their appointments.  The p-value for x equals 127 is 0.02094.

Determine the outcome of the hypothesis test, giving your conclusion in context.

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6a
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2 marks

Chase buys a board game which contains a six-sided dice.  He rolls the dice 150 times and obtains the number six on 15 occasions.  Chase wishes to test his belief that the dice is not fair.

(i)
State a suitable null hypothesis to test Chase’s belief.

(ii)
State a suitable alternative hypothesis for a two-tailed test.
6b
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3 marks

Given that straight P left parenthesis X less or equal than 15 right parenthesis equals 0.01452 space when  X tilde B left parenthesis 150 comma space 1 over 6 right parenthesis,  test Chase’s belief that the dice is not fair, using a 2% level of significance.

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7a
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2 marks

A test of the null hypothesis  straight H subscript 0 colon p equals 0.3  is carried out for the random variable X tilde B left parenthesis 25 comma p right parenthesis. The observed value of the test statistic is x equals 3. You are given the following probabilities:

straight P left parenthesis X less than 3 right parenthesis equals 0.00896

straight P left parenthesis X equals 3 right parenthesis equals 0.02428

straight P left parenthesis X less or equal than 3 right parenthesis equals 0.03324

Determine the outcome of the test, with reasons, when the alternative hypothesis is:

straight H subscript 1 colon p less than 0.3 with a 1% level of significance.

7b
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2 marks

straight H subscript 1 colon p not equal to 0.3 with a 5% level of significance.

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8a
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1 mark

A two-tailed test of the null hypothesis straight H subscript 0 colon p equals 0.23 is carried out for the random variable X tilde B left parenthesis 60 comma p right parenthesis

Write down the alternative hypothesis.

8b
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2 marks

One of the critical regions is X greater or equal than 20.  You are given the following probabilities:

straight P left parenthesis X less or equal than 8 right parenthesis equals 0.04603

straight P left parenthesis X less or equal than 9 right parenthesis equals 0.08932

text P end text left parenthesis X less or equal than 10 right parenthesis equals 0.15526

Given that a 10% level of significance is used, determine the other critical region. Give a reason for your answer by using a relevant probability.

8c
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2 marks

You are also given that straight P left parenthesis X greater or equal than 20 right parenthesis equals 0.04427..

Find the actual level of significance of this test.

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1a
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3 marks

In the context of hypothesis testing, explain the term:

(i)
critical region

(ii)
critical value
1b
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3 marks

The table below shows the probabilities for different values that X tilde B left parenthesis 40 comma 0.8 right parenthesis spacecan take:

x straight P left parenthesis X equals x right parenthesis
40 0.000133
39 0.001329
38 0.006480
37 0.020520
36 0.047452


A test of the null hypothesis  straight H subscript 0 colon p equals 0.8 against the alternative hypothesis straight H subscript 1 ∶ p greater than 0.8 is carried out for the random variable space X tilde B left parenthesis 40 comma p right parenthesis.

 Using a 5% level of significance, find the values of X which would lead to the rejection of the null hypothesis.

1c
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3 marks

A second test is carried out with the same null hypothesis against the alternative hypothesis H subscript 1 ∶ p not equal to 0.8.

Given that x equals 38 is a critical value, find the minimum level of significance for the test.

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2a
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5 marks

Meditest is a company manufacturing medical tests which are used to determine whether a patient has a certain illness.  Meditest claims that the tests are 95% accurate, however a particular hospital will only purchase the tests if they are more than 95% accurate.  Meditest test the accuracy of their product using a sample of 250 patients with the illness and agree on a 1% level of significance. They discover that the tests are accurate for 245 out of the 250 patients.

If X tilde B left parenthesis 250 comma 0.95 right parenthesis then straight P left parenthesis X equals 245 right parenthesis equals 0.008515 space and straight P left parenthesis X greater than 245 right parenthesis equals 0.004571.

Stating your hypotheses clearly, test whether Meditest’s product is more than 95% accurate using a 1% level of significance.

2b
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1 mark

Meditest notice that they would have had sufficient evidence to reject the null hypothesis using a 5% level of significance.  They change the level of significance from 1% to 5% and report to the hospital that their product is more than 95% accurate.

Comment on the validity of Meditest’s report to the hospital.

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3a
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4 marks

Frank is the owner of a factory which has recently opened near a school where Hilda is the headteacher.  Before the factory opened, the attendance rate at the school was good 90% of the time.  Hilda claims that the proportion of days when the attendance rate is good has decreased and she suspects this is due to the fumes from the factory making the children sick.  Frank disagrees and claims that the factory has made no difference to the attendance rate.  To test their claims a sample of 40 days is taken and on 32 days the attendance rate is good.

If space X tilde B left parenthesis 40 comma 0.9 right parenthesis space then:

straight P left parenthesis X less than 32 right parenthesis equals 0.015495

straight P left parenthesis X equals 32 right parenthesis equals 0.026407

straight P left parenthesis X greater than 32 right parenthesis equals 0.958098

Stating your hypotheses clearly, test Hilda’s claim using a 5% level of significance. Give your answer in context.

3b
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1 mark

Explain whether the outcome of the test supports Hilda’s suspicion.

3c
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3 marks

Stating your hypotheses clearly, test Frank’s claim using a 5% level of significance.

3d
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1 mark

Suggest a reason why Frank might want to use a two-tailed test.

3e
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2 marks

State, with a reason, whether a one-tailed test or a two-tailed test would have been more appropriate for this scenario.

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4a
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4 marks

Given that X tilde B left parenthesis 40 comma 0.3 right parenthesis then:

straight P left parenthesis X less or equal than 5 right parenthesis equals 0.008618

straight P left parenthesis X equals 6 right parenthesis equals k

straight P left parenthesis X equals 7 right parenthesis equals 0.031522

When a sample of size 40 is used to test straight H subscript 0 ∶ p equals 0.3 against straight H subscript 1 ∶ p less than 0.3, it is known that space x equals 6 space is the critical value using a 5% level of significance. Use the probabilities above to find upper and lower bounds for the value of k.

4b
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2 marks

When a sample of size 40 is used to test straight H subscript 0 ∶ p equals 0.3 against straight H subscript 1 ∶ p not equal to 0.3, it is known that x equals 6 spaceis one of the two critical values using a 5% level of significance. Use the probabilities above to find an improvement for one of the bounds for the value of k.

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5a
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2 marks

If X tilde B left parenthesis n comma p right parenthesis then straight P left parenthesis X equals 0 right parenthesis equals left parenthesis 1 minus p right parenthesis to the power of n and straight P left parenthesis X equals n right parenthesis equals p to the power of n.

A sample of size 30 is used to test the null hypothesis straight H subscript 0 ∶ p equals 0.9 against text H end text subscript 1 ∶ p greater than 0.9 the alternative hypothesis using a k% level of significance.

Given that there is at least one value that leads to the rejection of the null hypothesis, find the range of values for k.

5b
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3 marks

A sample of size 100 is used to test the null hypothesis  straight H subscript 0 ∶ p equals q against the alternative hypothesis straight H subscript 1 ∶ p less than q using a 5% level of significance.

Given that there are no critical values for this test, find the range of values for q.

5c
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4 marks

A sample of size m is used to test the null hypothesis straight H subscript 0 ∶ p equals 0.2 against the alternative hypothesis straight H subscript 1 ∶ p not equal to 0.2 using a 1% level of significance.

Given that there is exactly one critical region for this test, find the range of values for m.

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