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

Exam Questions

4 hours32 questions
1a
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4 marks

A random variable has distribution B left parenthesis n comma p right parenthesis. Anna uses a single observation of the random variable to carry out a hypothesis test.

Write down the conditions that must be met in order to model a random variable using the binomial distribution.

1b
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2 marks

Explain how the parameters n and p are used in the context of a hypothesis test.

1c
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1 mark

Anna is carrying out a two-tailed hypothesis test. Explain how what is being tested for in a two-tailed test differs from what is being tested for in a one-tailed test.

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2a
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3 marks

For the random variable X space tilde space B left parenthesis 20 comma 0.3 right parenthesis calculate

(i)
straight P left parenthesis X less or equal than 2 right parenthesis

(ii)
straight P left parenthesis X less than 3 right parenthesis
2b
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2 marks

A hypothesis test is to be carried out using an observation of the random variable X to test the hypotheses:

straight H subscript 0 ∶ p equals 0.3 space space space space space space space space space straight H subscript 1 ∶ p less than 0.3

Before carrying out the test, a significance level of 5% is chosen. The critical region is defined as X less or equal than 2.

(i)
Explain why the critical region is defined as X less or equal than 2.

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

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3a
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3 marks

Alina is running for student council president at her school.  She claims she has the support of 60% of students in the school.  A rival candidate, John, wants to test at the 10% level of significance whether Alina is overestimating her support.

In mathematical terms, the experiment John conducts uses an observation of the random variable  X space tilde space B left parenthesis 100 comma p right parenthesis space spaceto test the hypotheses:

straight H subscript 0 ∶ p equals 0.6 space space space space space space space space space space space straight H subscript 1 ∶ p less than 0.6

(i)
State how many people are in John’s sample and explain what the parameter p means in the context of this question.

(ii)
Explain why John has chosen this alternative hypothesis for his test.
3b
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4 marks

Assuming Alina’s claim is true, John calculates the following cumulative probabilities:

x 51 52 53 54 55
straight P left parenthesis X less or equal than x right parenthesis 0.04230 0.06379 0.09298 0.13109 0.17890


In John’s survey, 55 people say that they will support Alina.

(i)
Calculate straight P left parenthesis X equals 55 right parenthesis.

(ii)
State whether John should compare the value of straight P left parenthesis X less or equal than 55 right parenthesis or  straight P left parenthesis X equals 55 right parenthesis with his significance level of 10%.

(iii)
In the context of this question, write a conclusion for John’s test.
3c
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2 marks

Write down the critical value and the critical region for John’s test.

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4a
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3 marks

A hypothesis test at the 6% significance level is carried out on a coin using the following hypotheses:

straight H subscript 0 ∶ p equals 1 half space space space space space space space space straight H subscript 1 colon p not equal to 1 half

(i)
Give an example of what the parameter, p, could represent.

(ii)
In the context of your answer from part (i), explain what is meant by p equals begin inline style begin display style 1 half end style end style.

(iii)
This is a two-tailed test. Explain what should be done with the 6% significance level.
4b
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2 marks

A single observation x is to be taken from a binomial distribution X space tilde space B left parenthesis 100 comma p right parenthesis to test the hypotheses for the coin. One tail of the critical region is found to be  X less or equal than 6.

(i)
Using your knowledge of the binomial distribution B left parenthesis 100 comma 0.5 right parenthesis, write down the other tail of the critical region.

(ii)
Write down the set of values for x which would lead to the acceptance of the null hypothesis.

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5
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4 marks

In a school where all the teachers drink coffee in the staffroom, a headteacher wants to see if using a new brand of coffee in the staffroom improves teachers’ report writing punctuality.  Previously, 75% of teachers in her school would meet the deadline for writing reports.  After the new brand of coffee is introduced the headteacher takes a random sample of 15 teachers, and conducts an experiment to test whether the proportion of teachers meeting the deadline on the next set of reports has improved.  The test is conducted at the 5% significance level, using the following hypotheses:

straight H subscript 0 ∶ p equals 0.75 space space space space space space space straight H subscript 1 ∶ p greater than 0.75

(i)
Write down a suitable distribution for the random variable X, the number of teachers in the headteacher’s sample who meet the deadline.

(ii)
Calculate the probability that all the teachers in the sample would meet the deadline if the null hypothesis were true.

(iii)
By first calculating the probability that exactly 14 of the teachers would meet the deadline if the null hypothesis were true, find the critical value for the test.

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6a
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4 marks

A study of ladybirds in the UK found that 65% of all ladybirds are found to be of the seven-spot species.  Alex believes that more than 65% of the ladybirds in his garden are of the seven-spot species.  He conducts a hypothesis test at the 5% significance level by collecting a sample of 25 ladybirds from his garden and counting the number of them that are seven-spot ladybirds.

(i)
Alex uses the random variable X space tilde space B left parenthesis n comma p right parenthesis to represent the number of seven-spot ladybirds in his sample.  Explain what n and p represent in the context of Alex’ experiment.

(ii)
State an assumption Alex has made in order to use the distribution in part (a)(i).

(iii)
State suitable null and alternative hypotheses that Alex could use to test his belief that more than 65% of the ladybirds in his garden are seven-spot ladybirds.
6b
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3 marks

Alex finds that 21 out of the 25 ladybirds in his sample are seven-spot ladybirds.

(i)
The p-value for the observed test statistic is 0.03205. Write this in the form  P left parenthesis X greater or equal than a right parenthesis equals space b.
.
(ii)
Alex has not yet calculated the critical value for his hypothesis test.  Explain why he does not need to do this to come to a conclusion for his test.

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7a
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3 marks

The probability of a student in a primary school library returning his or her books on time had been found to be 0.35.  Joanna, the school librarian, has started a new incentive scheme and believes that more students are now returning their books on time because of it.  She conducts a hypothesis test using the null hypothesis  H subscript 0 colon p equals 0.35 to test her belief

(i)
State a suitable alternative hypothesis to test Joanna’s belief that more students are now returning their books on time.

(ii)
Write down the conditions under which Joanna could use a binomial probability distribution to model this problem.
7b
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2 marks

Joanna takes a random sample of 30 students who have checked out books and finds that under the new incentive scheme 15 of them return their books on time.  She calculates the following probabilities for the random variable  X space tilde space B left parenthesis 30 comma 0.35 right parenthesis :

straight P left parenthesis X equals 14 right parenthesis space equals space 0.06112

straight P left parenthesis X equals 15 right parenthesis equals space 0.03511

straight P left parenthesis X greater or equal than 16 right parenthesis equals space 0.03008

Write down the values of straight P left parenthesis X greater or equal than 15 right parenthesis and  straight P left parenthesis X greater or equal than 14 right parenthesis.

7c
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3 marks

Write a conclusion for the hypothesis test, in context, if Joanna had chosen a significance level of:

(i)
5%

(ii)
10%

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8a
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3 marks

The number of leaves on the grass in Danny’s garden on October 1st every year has a Poisson distribution with mean 280.

Given that Danny’s garden is 56 square metres,

(i)
Write down the distribution for the number of leaves per square metre in Danny’s garden on October 1st,
(ii)
Find the probability that in a randomly chosen square metre of Danny’s garden, there are less than 3 leaves.
8b
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2 marks

Danny’s neighbour has cut down a tree and Danny believes this will decrease the number of leaves in his garden this year. He is devising a hypothesis test to test his theory at the 10% significance level. On October 1st he will choose a square metre of his garden at random and count the number of leaves.

Write down suitable null and alternative hypotheses Danny can use for his test.

8c
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2 marks

Use your answer to part (a)(ii) to determine if finding 2 leaves will lead Danny to accept or reject his null hypothesis. Justify your answer.

8d
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3 marks

Determine the critical region for this hypothesis test and hence, write down the maximum number of leaves that will lead Danny to reject his null hypothesis.

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9a
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4 marks

A company releases a rapid testing kit for a particular virus and claims that the kit is correct 90% of the time. A medical professional believes this is false advertising and carries out a hypothesis test at the 1% significance level to test the company’s claim.

For each of the following statements, write down whether the hypothesis test has resulted in an error, and if so state whether it is a Type I or a Type II error.

(i)
The kit is correct 85% of the time. The medical professional rejects the null hypothesis in favour of the alternative hypothesis.
(ii)
The kit is correct 95% of the time. The medical professional accepts the null hypothesis.
(iii)
The kit is correct 90% of the time. The medical professional rejects the null hypothesis in favour of the alternative hypothesis.
(iv)
The kit is correct 90% of the time. The medical professional accepts the null hypothesis.
9b
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1 mark

Write down the greatest possible probability of the hypothesis test resulting in a Type I error.

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1a
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3 marks

A single observation is taken from a discrete random variable X tilde B left parenthesis 35 comma 0.4 right parenthesis  to test straight H subscript 0 colon p equals 0.4 against straight H subscript 1 colon p less than 0.4

Using a 5% level of significance, find the critical region for this test.

1b
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2 marks

The actual value for the observation was 4.

State a conclusion to the hypothesis test for this value, giving a reason for your answer.

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2a
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4 marks

Harry is using the random variable X tilde B left parenthesis 40 comma 0.15 right parenthesis  to test the hypotheses:

straight H subscript 0 colon p equals 0.15

straight H subscript 1 colon p not equal to 0.15

Harry states that the critical regions are X less or equal than 2 and X greater or equal than 11.

(i)
Given that P left parenthesis X greater or equal than 11 right parenthesis space equals space 0.029921, calculate the probability of incorrectly rejecting the null hypothesis.

(ii)
State which type of error is represented in part (a)(i).

(iii)
State, with a reason, the conclusion of Harry’s test given that the observed value is