A LevelMaths: Probability & Statistics 2CIEExam Questions3. Hypothesis Testing3.1 Hypothesis Testing

Hypothesis Testing (CIE A Level Maths: Probability & Statistics 2)

Exam Questions

3 hours30 questions
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1
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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)
‘a Type I error’,

(vi)
‘a Type II error’.

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2a
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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.

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2b
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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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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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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.  he 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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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, , 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.

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5b
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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 a Type II error could occur.

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6a
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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.  

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6b
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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.

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6c
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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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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.

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7b
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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. 

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7c
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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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8a
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For each of the following statements, write down whether an error has been made, and if so state whether it is a Type I or a Type II error.

(i)

H subscript 0 is true and H subscript 0 is accepted.

(ii)

H subscript 0 is true and H subscript 0 is rejected.

(iii)

H subscript 0 is not true and H subscript 0 is accepted.

(iv)

H subscript 0 is not true and H subscript 0 is rejected.

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8b
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Explain why the probability of a Type I error is usually just below the significance level.

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8c
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Describe how to calculate the probability of a Type II error.

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1a
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Explain what you understand by a critical region of a test statistic.

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1b
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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.

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1c
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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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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.

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2b
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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.

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2c
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(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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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.

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3b
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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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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.

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4b
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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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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.

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5b
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Given that for this hypothesis test the random variable to be used is X tilde B left parenthesis 160 comma p right parenthesis, describe in context what X represents.

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5c
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Out of the 160 patients used for the sample, 127 turned up for their appointments. Under the assumption that the null hypothesis is true, it is given that P left parenthesis X greater or equal than space 127 right parenthesis space equals space 0.02094.

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

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6a
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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.

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6b
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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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A test of the null hypothesis  straight H subscript 0 colon p equals 0.3  is carried out for the random variable