Perform one-tailed hypothesis test

5 A manufacturer produces titanium bicycle frames. The bicycle frames are tested before use and on average \(5 \%\) of them are found to be faulty. A cheaper manufacturing process is introduced and the manufacturer wishes to check whether the proportion of faulty bicycle frames has increased. A random sample of 18 bicycle frames is selected and it is found that 4 of them are faulty. Carry out a hypothesis test at the \(5 \%\) significance level to investigate whether the proportion of faulty bicycle frames has increased.

5 A researcher is investigating whether people can identify whether a glass of water they are given is bottled water or tap water. She suspects that people do no better than they would by guessing. Twenty people are selected at random; thirteen make a correct identification. She carries out a hypothesis test.

1. Explain why the null hypothesis should be \(p = 0.5\), where \(p\) represents the probability that a randomly selected person makes a correct identification.
2. Briefly explain why she uses an alternative hypothesis of \(p > 0.5\).
3. Complete the test at the \(5 \%\) significance level.

4 A television company believes that the proportion of adults who watched a certain programme is 0.14 . Out of a random sample of 22 adults, it is found that 2 watched the programme.

1. Carry out a significance test, at the \(10 \%\) level, to determine, on the basis of this sample, whether the television company is overestimating the proportion of adults who watched the programme.
2. The sample was selected randomly. State what properties of this method of sampling are needed to justify the use of the distribution used in your test.

3 An electronics company is developing a new sound system. The company claims that \(60 \%\) of potential buyers think that the system would be good value for money. In a random sample of 12 potential buyers, 4 thought that it would be good value for money. Test, at the 5\% significance level, whether the proportion claimed by the company is too high.

4 The proportion of commuters in a town who travel to work by train is 0.4 . Following the opening of a new station car park, a random sample of 16 commuters is obtained, and 11 of these travel to work by train. Test at the \(1 \%\) significance level whether there is evidence of an increase in the proportion of commuters in this town who travel to work by train.

5 A travel company finds from its records that \(40 \%\) of its customers book with travel agents. The company redesigns its website, and then carries out a survey of 10 randomly chosen customers. The result of the survey is that 1 of these customers booked with a travel agent.

1. Test at the \(5 \%\) significance level whether the percentage of customers who book with travel agents has decreased.
2. The managing director says that "Our redesigned website has resulted in a decrease in the percentage of our customers who book with travel agents." Comment on this statement.

3 It is known that on average one person in three prefers the colour of a certain object to be blue. In a psychological test, 12 randomly chosen people were seated in a room with blue walls, and asked to state independently which colour they preferred for the object. Seven of the 12 people said that they preferred blue. Carry out a significance test, at the \(5 \%\) level, of whether the statement "on average one person in three prefers the colour of the object to be blue" is true for people who are seated in a room with blue walls.

12 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. 12

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.
   [0pt] [6 marks] 12
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.
   [0pt] [4 marks]

1. Past records show that \(15 \%\) of customers at a shop buy chocolate. The shopkeeper believes that moving the chocolate closer to the till will increase the proportion of customers buying chocolate.

After moving the chocolate closer to the till, a random sample of 30 customers is taken and 8 of them are found to have bought chocolate. Julie carries out a hypothesis test, at the 5\% level of significance, to test the shopkeeper's belief.
Julie's hypothesis test is shown below.
\(\mathrm { H } _ { 0 } : p = 0.15\)
\(\mathrm { H } _ { 1 } : p \geqslant 0.15\)
Let \(X =\) the number of customers who buy chocolate.
\(X \sim \mathrm {~B} ( 30,0.15 )\)
\(\mathrm { P } ( X = 8 ) = 0.0420\)
\(0.0420 < 0.05\) so reject \(\mathrm { H } _ { 0 }\)
There is sufficient evidence to suggest that the proportion of customers buying chocolate has increased.

1. Identify the first two errors that Julie has made in her hypothesis test.
2. Explain whether or not these errors will affect the conclusion of her hypothesis test. Give a reason for your answer.
3. Find, using a 5\% level of significance, the critical region for a one-tailed test of the shopkeeper's belief. The probability in the tail should be less than 0.05
4. Find the actual level of significance of this test.

4. The random variable \(X \sim \mathrm {~B} ( 27,0.35 )\)

1. Find
   1. \(\mathrm { P } ( X = 10 )\)
   2. \(\mathrm { P } ( 12 \leqslant X < 15 )\) Historical records show that the proportion of defective items produced by a machine is 0.12 Following a maintenance service of the machine, a random sample of 60 items is taken and 3 defective items are found.
2. Carry out a suitable test to determine whether the proportion of defective items produced by the machine has decreased following the maintenance service. You should state your hypotheses clearly and use a \(5 \%\) level of significance.
3. Write down the \(p\)-value for your test in part (b)

12 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.

11 It is known that, under the standard treatment for a certain disease, \(9.7 \%\) of patients with the disease experience side effects within one year. In a trial of a new treatment, a random sample of 450 patients with this disease was selected and the number \(X\) who experienced side effects within one year was noted.

1. State one assumption needed in order to use a binomial model for \(X\). It was found that 51 of the 450 patients experienced side effects within one year.
2. Test, at the \(10 \%\) significance level, whether the proportion of patients experiencing side effects within one year is greater under the new treatment than under the standard treatment.

10 Some packets of a certain kind of biscuit contain a free gift. The manufacturer claims that the proportion of packets containing a free gift is 1 in 4 . Marisa suspects that this claim is not true, and that the true proportion is less than 1 in 4 . She chooses 20 packets at random and finds that exactly 1 contains the free gift.

1. Use a binomial model to test the manufacturer's claim, at the \(2.5 \%\) significance level. The packets are packed in boxes, with each box containing 40 packets. Marisa chooses three boxes at random and finds that one box contains 19 packets with the free gift and the other two boxes contain no packets with the free gift.
2. Give a reason why this suggests that the binomial model used in part (a) may not be appropriate.

11 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 \(\mathrm { H } _ { 0 } : p = 0.75\) and \(\mathrm { H } _ { 1 } : p > 0.75\). 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 \(\mathrm { P } ( X = 50 ) = 0.0407\), 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.
2. Carry out the test.
3. 1. State which mathematical model is used in the calculation in part (b), including the value(s) of any parameter(s).
   2. The random sample was chosen without replacement. Explain whether this invalidates the model used in part (b).

9 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.

8 According to the latest research there are 19.8 million male drivers and 16.2 million female drivers on the roads in the UK.

1. A driver in the UK is selected at random. Find the probability that the driver is male.
2. Calculate the probability that there are 7 female drivers in a random sample of 25 UK drivers. When driving in a built-up area, Rebecca exceeded the speed limit and was obliged to attend a speed awareness course. Her husband said "It's nearly always male drivers who are speeding." When Rebecca attends the course, she finds that there are 25 drivers, 7 of whom are female. You should assume that the drivers on the speed awareness course constitute a random sample of drivers caught speeding.
3. In this question you must show detailed reasoning. Conduct a hypothesis test to determine whether there is any evidence at the \(5 \%\) level to suggest that male drivers are more likely to exceed the speed limit than female drivers.
4. State a modelling assumption that is necessary in order to conduct the hypothesis test in part (c).

13 In a report published in October 2021 it is stated that \(37 \%\) of adults in the United Kingdom never exercise or play sport. A researcher believes that the true percentage is less than this. They decide to carry out a hypothesis test at the \(5 \%\) level to investigate the claim.

1. State the null and alternative hypotheses for their test.
2. Define the parameter for their test. In a random sample of 118 adults, they find that 35 of them never exercise or play sport.
3. Carry out the test.

10 A company operates trains. The company claims that \(92 \%\) of its trains arrive on time. You should assume that in a random sample of trains, they arrive on time independently of each other.

1. Assuming that \(92 \%\) of the company's trains arrive on time, find the probability that in a random sample of 30 trains operated by this company
   1. exactly 28 trains arrive on time,
   2. more than 27 trains arrive on time. A journalist believes that the percentage of trains operated by this company which arrive on time is lower than \(92 \%\).
2. To investigate the journalist's belief a hypothesis test will be carried out at the \(1 \%\) significance level. A random sample of 18 trains is selected. For this hypothesis test,
   • state the hypotheses,
   • find the critical region.

12 A survey conducted in 2021 showed that 10\% of British adults were vegetarians. A dietitian believes that the proportion of British adults who are vegetarians may have changed, so decides to conduct a hypothesis test at the \(5 \%\) level of significance. In a random sample of 112 adults, the dietitian finds that there are 19 vegetarians. Carry out the hypothesis test to determine whether there is any evidence to support the dietitian’s belief.

2. Bill owns a restaurant. Over the next four weeks Bill decides to carry out a sample survey to obtain the customers’ opinions.

1. Suggest a suitable sampling frame for the sample survey.
2. Identify the sampling units.
3. Give one advantage and one disadvantage of taking a census rather than a sample survey. Bill believes that only \(30 \%\) of customers would like a greater choice on the menu. He takes a random sample of 50 customers and finds that 20 of them would like a greater choice on the menu.
4. Test, at the \(5 \%\) significance level, whether or not the percentage of customers who would like a greater choice on the menu is more than Bill believes. State your hypotheses clearly.

6. The Headteacher of a school claims that \(30 \%\) of parents do not support a new curriculum. In a survey of 20 randomly selected parents, the number, \(X\), who do not support the new curriculum is recorded. Assuming that the Headteacher's claim is correct, find

1. the probability that \(X = 5\)
2. the mean and the standard deviation of \(X\) The Director of Studies believes that the proportion of parents who do not support the new curriculum is greater than \(30 \%\). Given that in the survey of 20 parents 8 do not support the new curriculum,
3. test, at the \(5 \%\) level of significance, the Director of Studies' belief. State your hypotheses clearly. The teachers believe that the sample in the original survey was biased and claim that only \(25 \%\) of the parents are in support of the new curriculum. A second random sample, of size \(2 n\), is taken and exactly half of this sample supports the new curriculum. A test is carried out at a 10\% level of significance of the teachers' belief using this sample of size \(2 n\) Using the hypotheses \(\mathrm { H } _ { 0 } : p = 0.25\) and \(\mathrm { H } _ { 1 } : p > 0.25\)
4. find the minimum value of \(n\) for which the outcome of the test is that the teachers' belief is rejected.

1. At a shop, past figures show that \(35 \%\) of customers pay by credit card. Following the shop’s decision to no longer charge a fee for using a credit card, a random sample of 20 customers is taken and 11 are found to have paid by credit card.

Hadi believes that the proportion of customers paying by credit card is now greater than 35\%

1. Test Hadi's belief at the \(5 \%\) level of significance. State your hypotheses clearly. For a random sample of 20 customers,
2. show that 11 lies less than 2 standard deviations above the mean number of customers paying by credit card.
   You may assume that \(35 \%\) is the true proportion of customers who pay by credit card.

1. Superbounce is a manufacturer of tennis balls.

It knows from past records that 10\% of its tennis balls fail a bounce test.

1. Find the probability that from a random sample of 10 of these tennis balls
   1. at least 4 fail the bounce test
   2. more than 1 but fewer than 5 fail the bounce test. The managing director makes changes to the production process and claims that these changes will reduce the probability of its tennis balls failing the bounce test. After the changes were made a random sample of 50 of the tennis balls were tested and it was found that 2 failed the bounce test.
2. Test, at the \(5 \%\) significance level, whether or not this result supports the managing director's claim. In a second random sample of \(n\) tennis balls it was found that none failed the bounce test. As a result of this sample, the managing director's claim is supported at the 1\% significance level.
3. Find the smallest possible value of \(n\)

6. The Headteacher of a school claims that \(30 \%\) of parents do not support a new curriculum. In a survey of 20 randomly selected parents, the number, \(X\), who do not support the new curriculum is recorded. Assuming that the Headteacher's claim is correct, find

1. the probability that \(X = 5\)
2. the mean and the standard deviation of \(X\) The Director of Studies believes that the proportion of parents who do not support the new curriculum is greater than \(30 \%\). Given that in the survey of 20 parents 8 do not support the new curriculum,
3. test, at the \(5 \%\) level of significance, the Director of Studies' belief. State your hypotheses clearly. The teachers believe that the sample in the original survey was biased and claim that only \(25 \%\) of the parents are in support of the new curriculum. A second random sample, of size \(2 n\), is taken and exactly half of this sample supports the new curriculum. A test is carried out at a \(10 \%\) level of significance of the teachers' belief using this sample of size \(2 n\) Using the hypotheses \(\mathrm { H } _ { 0 } : p = 0.25\) and \(\mathrm { H } _ { 1 } : p > 0.25\)
4. find the minimum value of \(n\) for which the outcome of the test is that the teachers' belief is rejected.