Two-tailed test critical region

A single observation \(x\) is to be taken from a Binomial distribution B(20, \(p\)). This observation is used to test \(H_0 : p = 0.3\) against \(H_1 : p \neq 0.3\)

1. Using a 5\% level of significance, find the critical region for this test. The probability of rejecting either tail should be as close as possible to 2.5\%. [3]
2. State the actual significance level of this test. [2]

The actual value of \(x\) obtained is 3.

1. State a conclusion that can be drawn based on this value giving a reason for your answer. [2]

Past records show that 20\% of customers who buy crisps from a large supermarket buy them in single packets. During a particular day a random sample of 25 customers who had bought crisps was taken and 2 of them had bought them in single packets.

1. Use these data to test, at the 5\% level of significance, whether or not the percentage of customers who bought crisps in single packets that day was lower than usual. State your hypotheses clearly. [6]

At the same supermarket, the manager thinks that the probability of a customer buying a bumper pack of crisps is 0.03. To test whether or not this hypothesis is true the manager decides to take a random sample of 300 customers.

1. Stating your hypotheses clearly, find the critical region to enable the manager to test whether or not there is evidence that the probability is different from 0.03. The probability for each tail of the region should be as close as possible to 2.5\%. [6]
2. Write down the significance level of this test. [1]

It is known from past records that 1 in 5 bowls produced in a pottery have minor defects. To monitor production a random sample of 25 bowls was taken and the number of such bowls with defects was recorded.

1. Using a 5\% level of significance, find critical regions for a two-tailed test of the hypothesis that 1 in 5 bowls have defects. The probability of rejecting, in either tail, should be as close to 2.5\% as possible. [6]
2. State the actual significance level of the above test. [1]

At a later date, a random sample of 20 bowls was taken and 2 of them were found to have defects.

1. Test, at the 10\% level of significance, whether or not there is evidence that the proportion of bowls with defects has decreased. State your hypotheses clearly. [7]

A company claims that a quarter of the bolts sent to them are faulty. To test this claim the number of faulty bolts in a random sample of 50 is recorded.

1. Give two reasons why a binomial distribution may be a suitable model for the number of faulty bolts in the sample. [2]
2. Using a 5\% significance level, find the critical region for a two-tailed test of the hypothesis that the probability of a bolt being faulty is \(\frac{1}{4}\). The probability of rejection in either tail should be as close as possible to 0.025 [3]
3. Find the actual significance level of this test. [2]

In the sample of 50 the actual number of faulty bolts was 8.

1. Comment on the company's claim in the light of this value. Justify your answer. [2]

The machine making the bolts was reset and another sample of 50 bolts was taken. Only 5 were found to be faulty.

1. Test at the 1\% level of significance whether or not the probability of a faulty bolt has decreased. State your hypotheses clearly. [6]

In Manuel's restaurant the probability of a customer asking for a vegetarian meal is 0.30. During one particular day in a random sample of 20 customers at the restaurant 3 ordered a vegetarian meal.

1. Stating your hypotheses clearly, test, at the 5\% level of significance, whether or not the proportion of vegetarian meals ordered that day is unusually low. [5]

Manuel's chef believes that the probability of a customer ordering a vegetarian meal is 0.10. The chef proposes to take a random sample of 100 customers to test whether or not there is evidence that the proportion of vegetarian meals ordered is different from 0.10.

1. Stating your hypotheses clearly, use a suitable approximation to find the critical region for this test. The probability for each tail of the region should be as close as possible to 2.5\%. [6]
2. State the significance level of this test giving your answer to 2 significant figures. [1]

A teacher wants to investigate the sports played by students at her school in their free time. She decides to ask a random sample of 120 pupils to complete a short questionnaire.

1. Give two reasons why the teacher might choose to use a sample survey rather than a census. [2 marks]
2. Suggest a suitable sampling frame that she could use. [1 mark]

The teacher believes that 1 in 20 of the students play tennis in their free time. She uses the data collected from her sample to test if the proportion is different from this.

1. Using a suitable approximation and stating the hypotheses that she should use, find the critical region for this test. The probability for each tail of the region should be as close as possible to 5\%. [6 marks]
2. State the significance level of this test. [1 mark]

It is known from historical data that 15% of the residents of a town buy the local weekly newspaper, 'Local News'. A new free weekly paper is introduced into the town. The owners of 'Local News' are interested to know whether the introduction of the free newspaper has changed the proportion of residents who buy their paper. In a random sample of 50 residents of the town taken after the free newspaper was introduced, it was found that 3 of them purchased 'Local News' regularly. Investigate, at the 5% significance level, whether this sample provides evidence that the proportion of local residents who buy 'Local News' has changed. [6 marks]

Suzanne is a member of a sports club. For each sport she competes in, she wins half of the matches.

1. After buying a new tennis racket Suzanne plays 10 matches and wins 7 of them. Investigate, at the 10% level of significance, whether Suzanne's new racket has made a difference to the probability of her winning a match. [7 marks]
2. After buying a new squash racket, Suzanne plays 20 matches. Find the minimum number of matches she must win for her to conclude, at the 10% level of significance, that the new racket has improved her performance. [5 marks]

It is known that 80% of all diesel cars registered in 2017 had carbon monoxide (CO) emissions less than 0.3 g/km. Talat decides to investigate whether the proportion of diesel cars registered in 2022 with CO emissions less than 0.3 g/km has **changed**. Talat will carry out a hypothesis test at the 10% significance level on a random sample of 25 diesel cars registered in 2022.

1. 1. State suitable null and alternative hypotheses for Talat's test. [1 mark]
   2. Using a 10% level of significance, find the critical region for Talat's test. [5 marks]
   3. In his random sample, Talat finds 18 cars with CO emissions less than 0.3 g/km. State Talat's conclusion in context. [1 mark]
2. Talat now wants to use his random sample of 25 diesel cars, registered in 2022, to investigate whether the proportion of diesel cars in England with CO emissions more than 0.5 g/km has changed from the proportion given by the Large Data Set. Using your knowledge of the Large Data Set, give **two** reasons why it is not possible for Talat to do this. [2 marks]