2.05b Hypothesis test for binomial proportion

A train company claims that the probability \(p\) of one of its trains arriving late is 10\%. A regular traveller on the company's trains believes that the probability is greater than 10\% and decides to test this by randomly selecting 12 trains and recording the number \(X\) of trains that were late. The traveller sets up the hypotheses H\(_0\): \(p = 0.1\) and H\(_1\): \(p > 0.1\) and accepts the null hypothesis if \(x \leq 2\).

1. Find the size of the test. [1]
2. Show that the power function of the test is $$1 - (1 - p)^{10}(1 + 10p + 55p^2).$$ [4]
3. Calculate the power of the test when
   1. \(p = 0.2\),
   2. \(p = 0.6\). [3]
4. Comment on your results from part (c). [1]

Pierre is a chef. He claims that 90% of his customers are satisfied with his cooking. Yvette suspects that Pierre is over-confident about the level of satisfaction amongst his customers. She talks to a random sample of 15 of Pierre's customers, and finds that 11 customers say that they are satisfied. She then performs a hypothesis test. Carry out the test at the 5% significance level. [7]

The random variable \(Y\) has the distribution \(\text{N}(\mu, \sigma^2)\).

1. Find \(\text{P}(Y > \mu - \sigma)\). [1]
2. Given that \(\text{P}(Y > 45) = 0.2\) and \(\text{P}(Y < 25) = 0.3\), determine the values of \(\mu\) and \(\sigma\). [6]

The random variables \(U\) and \(V\) have the distributions \(\text{N}(10, 4)\) and \(\text{N}(12, 9)\) respectively.

1. It is given that \(\text{P}(U < b) = \text{P}(V > c)\), where \(b > 10\) and \(c < 12\). Determine \(b\) in terms of \(c\). [2]

Martin grows cucumbers from seed. In the past, he has found that 70% of all seeds successfully germinate and grow into cucumber plants. He decides to try out a new brand of seed. The producer of this brand claims that these seeds are more likely to successfully germinate than other brands of seeds. Martin sows 20 of this new brand of seed and 18 successfully germinate. Carry out a hypothesis test at the 5% level of significance to investigate the producer's claim. [7 marks]

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]

It is believed that 25% of the customers at a bakery buy a loaf of bread. In an attempt to increase this proportion, the manager of the bakery provided free samples for the customers to taste. To decide whether providing free samples had been effective, a large random sample of customers leaving the bakery were asked whether they had purchased a loaf of bread. A hypothesis test at the 5% significance level was carried out on the data collected. The test statistic calculated was found to be in the critical region.

1. State the Null and Alternative hypotheses for this test. [1 mark]
2. State, in context, the conclusion to this test. [2 marks]

The proportion of vegans in a city is thought to be 8% The owner of an organic food café in this city believes that the proportion of their customers who are vegan is greater than 8% To test this belief, a random sample of 50 customers at the café were interviewed and it was found that 7 of them were vegan. Investigate, at the 5% level, whether this sample supports the owner's belief. [5 marks]

Ellie, a statistics student, read a newspaper article that stated that 20 per cent of students eat at least five portions of fruit and vegetables every day. Ellie suggests that the number of people who eat at least five portions of fruit and vegetables every day, in a sample of size \(n\), can be modelled by the binomial distribution B(\(n\), 0.20).

1. There are 10 students in Ellie's statistics class. Using the distributional model suggested by Ellie, find the probability that, of the students in her class:
   1. two or fewer eat at least five portions of fruit and vegetables every day; [1 mark]
   2. at least one but fewer than four eat at least five portions of fruit and vegetables every day; [2 marks]
2. Ellie's teacher, Declan, believes that more than 20 per cent of students eat at least five portions of fruit and vegetables every day. Declan asks the 25 students in his other statistics classes and 8 of them say that they eat at least five portions of fruit and vegetables every day.
   1. Name the sampling method used by Declan. [1 mark]
   2. Describe one weakness of this sampling method. [1 mark]
   3. Assuming that these 25 students may be considered to be a random sample, carry out a hypothesis test at the 5\% significance level to investigate whether Declan's belief is supported by this evidence. [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]

Tiana is a quality controller in a clothes factory. She checks for four possible types of defects in shirts. Of the shirts with defects, the proportion of each type of defect is as shown in the table below.

Type of defect	Colour	Fabric	Sewing	Sizing
Probability	0.25	0.30	0.40	0.05

Shirts with defects are packed in boxes of 30 at random.

1. Find the probability that:
   1. a box contains exactly 5 shirts with a colour defect [2 marks]
   2. a box contains fewer than 15 shirts with a sewing defect [2 marks]
   3. a box contains at least 20 shirts which do not have a fabric defect. [3 marks]
2. Tiana wants to investigate the proportion, \(p\), of defective shirts with a fabric defect. She wishes to test the hypotheses H\(_0\): \(p = 0.3\) H\(_1\): \(p < 0.3\) She takes a random sample of 60 shirts with a defect and finds that \(x\) of them have a fabric defect.
   1. Using a 5\% level of significance, find the critical region for \(x\). [5 marks]
   2. In her sample she finds 13 shirts with a fabric defect. Complete the test stating her conclusion in context. [2 marks]

James is playing a mathematical game on his computer. The probability that he wins is 0.6 As part of an online tournament, James plays the game 10 times. Let \(Y\) be the number of games that James wins.

1. State two assumptions, in context, for \(Y\) to be modelled as \(B(10, 0.6)\) [2 marks]
2. Find \(P(Y = 4)\) [1 mark]
3. Find \(P(Y \geq 4)\) [2 marks]
4. After practising the game, James claims that he has increased his probability of winning the game. In a random sample of 15 subsequent games, he wins 12 of them. Test at a 5% significance level whether James's claim is correct. [6 marks]

A bank runs a campaign to promote Internet banking accounts to their customers. Before the campaign, 42% of their customers had an Internet banking account. One week after the campaign started, 35 customers were surveyed at random and 18 of them were found to have registered for an Internet banking account. Using a binomial distribution, carry out a hypothesis test at the 10% significance level to investigate the claim that, since the campaign, there has been an increase in the proportion of customers registered for an Internet banking account. [6 marks]

A council found that 70% of its new local businesses made a profit in their first year. The council introduced an incentive scheme for its residents to encourage the use of new local businesses. At the end of the scheme, a random sample of 25 new local businesses was selected and it was found that 21 of them had made a profit in their first year. Using a binomial distribution, investigate, at the 2.5% level of significance, whether there is evidence of an increase in the proportion of new local businesses making a profit in their first year. [6 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]

During the 2006 Christmas holiday, John, a maths teacher, realised that he had fallen ill during 65% of the Christmas holidays since he had started teaching. In January 2007, he increased his weekly exercise to try to improve his health. For the next 7 years, he only fell ill during 2 Christmas holidays.

1. Using a binomial distribution, investigate, at the 5% level of significance, whether there is evidence that John's rate of illness during the Christmas holidays had decreased since increasing his weekly exercise. [6 marks]
2. State two assumptions, regarding illness during the Christmas holidays, that are necessary for the distribution you have used in part (a) to be valid. For each assumption, comment, in context, on whether it is likely to be correct. [4 marks]

It is known that 20% of plants of a certain type suffer from a fungal disease, when grown under normal conditions. Some plants of this type are grown using a new method. A random sample of 250 of these plants is chosen, and it is found that 36 suffer from the disease. Test, at the 2% significance level, whether there is evidence that the new method reduces the proportion of plants which suffer from the disease. [7]

Casey and Riley attend a large school. They are discussing the music preferences of the students at their school. Casey believes that the favourite band of 75% of the students is Blue Rocking. Riley believes that the true figure is greater than 75%. They plan to carry out a hypothesis test at the 5% significance level, using the hypotheses \(H_0: p = 0.75\) and \(H_1: p > 0.75\). They choose a random sample of 60 students from the school, and note the number, \(X\), who say that their favourite band is Blue Rocking. They find that \(X = 50\).

1. Assuming the null hypothesis to be true, Riley correctly calculates that \(P(X = 50) = 0.0407\), correct to 3 significant figures. Riley says that, because this value is less than 0.05, the null hypothesis should be rejected. Explain why this statement is incorrect. [1]
2. Carry out the test. [5]
3. 1. State which mathematical model is used in the calculation in part (b), including the value(s) of any parameter(s). [1]
   2. The random sample was chosen without replacement. Explain whether this invalidates the model used in part (b). [1]

In this question you must show detailed reasoning. Research showed that in May 2017 62% of adults over 65 years of age in the UK used a certain online social media platform. Later in 2017 it was believed that this proportion had increased. In December 2017 a random sample of 59 adults over 65 years of age in the UK was collected. It was found that 46 of the 59 adults used this online social media platform. Use a suitable hypothesis test to determine whether there is evidence at the 1% level to suggest that the proportion of adults over 65 in the UK who used this online social media platform had increased from May 2017 to December 2017. [7]

Records from the 1950s showed that 35\% of human babies were born without wisdom teeth. It is believed that as part of the evolutionary process more babies are now born without wisdom teeth. In a random sample of 140 babies, collected in 2020, a researcher found that 61 were born without wisdom teeth. The researcher made the following statement. ``This shows that the percentage of babies born without wisdom teeth has increased from 35\%.''

1. Explain whether this statement can be fully justified. [1]
2. Conduct a hypothesis test at the 5\% level to determine whether there is any evidence to suggest that more than 35\% of babies are now born without wisdom teeth. [7]

Edward can correctly identify 20% of types of wild flower. He studies some books to see if he can improve how often he can correctly identify types of wild flower. He collects a random sample of 10 types of wild flower in order to test whether or not he has improved.

1. 1. Write suitable hypotheses for this test.
   2. State a suitable test statistic that he could use. [2]
2. Using a 5% level of significance, find the critical region for this test. [3]
3. State the probability of a Type I error for this test and explain what it means in this context. [2]
4. Edward correctly identifies 4 of the 10 types of wild flower he collected. What conclusion should Edward reach? [2]

A company produces sweets of varying colours. The company claims that the proportion of blue sweets is 13·6%. A consumer believes that the true proportion is less than this. In order to test this belief, the consumer collects a random sample of 80 sweets.

1. State suitable hypotheses for the test. [1]
2. 1. Determine the critical region if the test is to be carried out at a significance level as close as possible to, but not exceeding, 5%.
   2. Given that there are 6 blue sweets in the sample of 80, complete the significance test. [5]
3. Suppose the proportion of blue sweets claimed by the company is correct. The consumer conducts the sampling and testing process on a further 20 occasions, using the sample size of 80 each time. What is the expected number of these occasions on which the consumer would reach the incorrect conclusion? [2]
4. Now suppose that the proportion of blue sweets is 7%. Find the probability of a Type II error. Interpret your answer in context. [3]

Dewi, a candidate in an election, believes that 45% of the electorate intend to vote for him. His agent, however, believes that the support for him is less than this. Given that \(p\) denotes the proportion of the electorate intending to vote for Dewi,

1. state hypotheses to be used to resolve this difference of opinion. [1]

They decide to question a random sample of 60 electors. They define the critical region to be \(X \leq 20\), where \(X\) denotes the number in the sample intending to vote for Dewi.

1. 1. Determine the significance level of this critical region.
   2. If in fact \(p\) is actually 0.35, calculate the probability of a Type II error.
   3. Explain in context the meaning of a Type II error.
   4. Explain briefly why this test is unsatisfactory. How could it be improved while keeping approximately the same significance level? [8]

Tony runs a pie stand that sells two types of pie outside a football ground. He wants to estimate the probability that a customer will buy a steak pie rather than a vegetable pie. He conducts a survey by randomly selecting customers and recording their choice of pie. When he feels he has enough data, he notes that 55 customers bought steak pies and 25 bought vegetable pies.

1. Calculate an approximate 90\% confidence interval for \(p\), the probability that a randomly selected customer buys a steak pie. [6]
2. Suppose that Tony carries out 50 such surveys and calculates 90\% confidence intervals for each survey. Determine the expected number of these confidence intervals that would contain the true value of \(p\). [1]

Tiana is a quality controller in a clothes factory. She checks for four possible types of defects in shirts. Of the shirts with defects, the proportion of each type of defect is as shown in the table below. Tiana wants to investigate the proportion, \(p\), of defective shirts with a fabric defect. She wishes to test the hypotheses $$H_0 : p = 0.3$$ $$H_1 : p < 0.3$$ She takes a random sample of 60 shirts with a defect and finds that \(x\) of them have a fabric defect.

1. Using a 5% level of significance, find the critical region for \(x\). [5 marks]
2. In her sample she finds 13 shirts with a fabric defect. Complete the test stating her conclusion in context. [2 marks]

A firm claims that no more than 2\% of their packets of sugar are underweight. A market researcher believes that the actual proportion is greater than 2\%. In order to test the firm's claim, the researcher weighs a random sample of 600 packets and carries out a hypothesis test, at the 5\% significance level, using the null hypothesis \(p = 0.02\).

1. Given that the researcher's null hypothesis is correct, determine the probability that the researcher will conclude that the firm's claim is incorrect. [5]
2. The researcher finds that 18 out of the 600 packets are underweight. A colleague says "18 out of 600 is 3\%, so there is evidence that the actual proportion of underweight bags is greater than 2\%." Criticise this statement. [1]