A LevelMathsEdexcelStatisticsExam QuestionsHypothesis TestingIntroduction to Hypothesis Testing

Introduction to Hypothesis Testing (Edexcel A Level Maths: Statistics): Exam Questions

Exam code: 9MA0

2 hours20 questions
Easy
Medium
Hard
Very Hard
1a
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.

How did you do?

1b
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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.       

How did you do?

1c
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.

How did you do?

2a
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.

How did you do?

2b
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).

How did you do?

3
2 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 \text{H}_{0}: p = \dfrac{1}{3} to test her belief at the 10% significance level.

The teacher wishes to test whether 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.

How did you do?

4a
3 marks

A spinner has four sections labelled A, B, C and D. The probability that the spin lands on section A is p.

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

\text{H}_{0}: p = \dfrac{1}{4}, \; \text{H}_{1}: p \neq \dfrac{1}{4}

(i) Describe, in context, what the hypotheses are testing.

(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.

How did you do?

4b
2 marks

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

(i) The critical regions for this test are given as X \leq a and X \geq b. Write down the values of a and b.

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

How did you do?

5a
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:

\text{H}_{0}: p = 0.2, \; \text{H}_{1}: p \neq 0.2

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

(ii) State whether this is a one-tailed or a two-tailed test.

How did you do?

5b
2 marks

Owen observes a random sample of 100 people at the national park and finds that 14 of them left their litter behind. He calculates that if \text{H}_{0} were true, then the probability of 14 or fewer people leaving litter would be 0.08044.

With reference to the hypotheses above, state with a reason whether or not Owen should reject his null hypothesis.

How did you do?

5c
3 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:

\text{H}_{0}: p = 0.2, \; \text{H}_{1}: p < 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.

How did you do?

1a
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.

How did you do?

1b
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.

How did you do?

2a
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.

How did you do?

2b
2 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.

How did you do?

3a
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.

How did you do?

3b
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 \text{P}(X \leq 2) = 0.01127 when X \sim \text{B}(100, 0.08), determine the outcome of the hypothesis test using a 1% level of significance. Give your conclusion in context.

How did you do?

4a
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.

How did you do?

4b
1 mark

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

How did you do?

4c
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.

How did you do?

5a
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.

How did you do?

5b
3 marks

Given that \text{P}(X \leq 15) = 0.01452 when X \sim \text{B}\left(150, \dfrac{1}{6}\right), test Chase's belief that the dice is not fair, using a 2% level of significance.

How did you do?

6a
2 marks

A test of the null hypothesis \text{H}_{0}: p = 0.3 is carried out for the random variable X \sim \text{B}(25, p). The observed value of the test statistic is x = 3. You are given the following probabilities:

\text{P}(X < 3) = 0.00896

\text{P}(X = 3) = 0.02428

\text{P}(X \leq 3) = 0.03324

Determine the outcome of the test, with reasons, when the alternative hypothesis is \text{H}_{1}: p < 0.3 with a 1% level of significance.

How did you do?

6b
2 marks

Determine the outcome of the test, with reasons, when the alternative hypothesis is \text{H}_{1}: p \neq 0.3 with a 5% level of significance.

How did you do?

1a
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 \sim \text{B}(100, 0.5) then:

\text{P}(X < 42) = 0.0443

\text{P}(X \leq 42) = 0.0666

\text{P}(X = 42) = 0.0223

\text{P}(X \geq 42) = 0.9557

\text{P}(X > 42) = 0.9334

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

How did you do?

1b
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.

How did you do?

2
3 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 \sim \text{B}\left(80, \dfrac{3}{4}\right) the following probabilities are given:

\text{P}(X < 65) = 0.8792

\text{P}(X \leq 65) = 0.9260

\text{P}(X = 65) = 0.0468

\text{P}(X \geq 65) = 0.1208

\text{P}(X > 65) = 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.

How did you do?

3a
1 mark

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

How did you do?

3b
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2 marks

A test of the null hypothesis \text{H}_{0}: p = 0.2 against the alternative hypothesis \text{H}_{1}: p < 0.2 is carried out for the random variable X \sim \text{B}(n, p).

The table below shows the probabilities for different values that X \sim \text{B}(n, 0.2) can take:

x

\text{P}(X = x)

0

0.000406

1

0.003549

2

0.015085

3

0.041484

4

0.082968

Calculate the p-value for the test statistic x = 3.

How did you do?

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.

How did you do?

4a
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:

\text{H}_{0}: p = 0.6, \text{H}_{1}: p \geq 0.6

Nate's solution:

\text{H}_{0}: p = 0.6, \text{H}_{1}: p > \dfrac{24}{30} = 0.8

Let X be the number of games won, X \sim \text{B}(30, 0.6).

\text{P}(X = 24) = 0.0115.

0.0115 < 0.05 so do not reject \text{H}_{0}

Let X be the number of games won, X \sim \text{B}(30, 0.6).

\text{P}(X > 24) = 0.0057.

0.0057 < 0.05 so reject \text{H}_{0}

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

Identify and explain the three mistakes made by Gertrude.

How did you do?

4b
2 marks

Identify and explain the two mistakes made by Nate.

How did you do?

5a
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.

How did you do?

5b
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2 marks

Calculate the probability of incorrectly rejecting the null hypothesis.

How did you do?

5c
1 mark

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

How did you do?

6
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3 marks

QualiCheck is a company manufacturing sensors used to detect faulty items on a production line. QualiCheck claims that the sensors are 95% accurate, however a large factory will only purchase the sensors if they are more than 95% accurate. QualiCheck tests the accuracy of their sensors using a sample of 250 items and agrees on a 1% level of significance. They find that the sensors are accurate for 245 out of the 250 items.

If X \sim \text{B}(250, \; 0.95) then \text{P}(X = 245) = 0.008515 and \text{P}(X > 245) = 0.004571.

Stating your hypotheses clearly, test whether QualiCheck's sensors are more than 95% accurate using a 1% level of significance.

How did you do?

7a
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4 marks

Frank is the operations manager of a bus company, and Hilda is a regular commuter. Before a major roadworks project began on the route, the morning bus arrived on time 90% of the time. Hilda claims that the proportion of days on which the bus is on time has decreased since the roadworks began, and she suspects this is because of the roadworks. Frank disagrees and claims that the roadworks have made no difference to the punctuality. To test their claims a sample of 40 days is taken, and on 32 of these days the bus arrived on time.

If X \sim \text{B}(40, \; 0.9) then:

\text{P}(X < 32) = 0.015495

\text{P}(X = 32) = 0.026407

\text{P}(X > 32) = 0.958098

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

How did you do?

7b
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3 marks

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

How did you do?

1a
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3 marks

Given that X tilde text B end text left parenthesis 40 comma space q right parenthesis then:

\text{P}(X \leq 5) = 0.008618

\text{P}(X = 6) = k

\text{P}(X = 7) = 0.031522

When a sample of size 40 is used to test text H end text subscript 0 colon p equals q against text H end text subscript 1 colon p less than q, it is known that x = 6 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.

How did you do?

1b
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2 marks

When a sample of size 40 is used to test text H end text subscript 0 colon p equals q against text H end text subscript 1 colon p not equal to q, it is known that x = 6 is 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.

How did you do?

2a
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2 marks

If X \sim \text{B}(n, \; p) then \text{P}(X = 0) = (1 - p)^{n} and \text{P}(X = n) = p^{n}.

A sample of size 30 is used to test the null hypothesis \text{H}_{0} : p = 0.9 against the alternative hypothesis \text{H}_{1} : p > 0.9 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.

How did you do?

2b
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3 marks

A sample of size 100 is used to test the null hypothesis \text{H}_{0} : p = q against the alternative hypothesis \text{H}_{1} : p < q using a 5% level of significance.

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

How did you do?

2c
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3 marks

A sample of size m is used to test the null hypothesis \text{H}_{0} : p = 0.2 against the alternative hypothesis \text{H}_{1} : p \neq 0.2 using a 1% level of significance. The probability of rejection in each tail is less than 0.005.

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

How did you do?