Perform one-tailed hypothesis test

2 Before attending a basketball course, a player found that \(60 \%\) of his shots made a score. After attending the course the player claimed he had improved. In his next game he tried 12 shots and scored in 10 of them. Assuming shots to be independent, test this claim at the \(10 \%\) significance level.

2 Arvind uses an ordinary fair 6-sided die to play a game. He believes he has a system to predict the score when the die is thrown. Before each throw of the die, he writes down what he thinks the score will be. He claims that he can write the correct score more often than he would if he were just guessing. His friend Laxmi tests his claim by asking him to write down the score before each of 15 throws of the die. Arvind writes the correct score on exactly 5 out of 15 throws. Test Arvind's claim at the \(10 \%\) significance level.

7 Every July, as part of a research project, Rita collects data about sightings of a particular kind of bird. Each day in July she notes whether she sees this kind of bird or not, and she records the number \(X\) of days on which she sees it. She models the distribution of \(X\) by \(\mathrm { B } ( 31 , p )\), where \(p\) is the probability of seeing this kind of bird on a randomly chosen day in July. Data from previous years suggests that \(p = 0.3\), but in 2022 Rita suspected that the value of \(p\) had been reduced. She decided to carry out a hypothesis test. In July 2022, she saw this kind of bird on 4 days.

1. Use the binomial distribution to test at the \(5 \%\) significance level whether Rita's suspicion is justified.
   In July 2023, she noted the value of \(X\) and carried out another test at the \(5 \%\) significance level using the same hypotheses.
2. Calculate the probability of a Type I error.
   Rita models the number of sightings, \(Y\), per year of a different, very rare, kind of bird by the distribution \(B ( 365,0.01 )\).
3. 1. Use a suitable approximating distribution to find \(\mathrm { P } ( Y = 4 )\).
   2. Justify your approximating distribution in this context.
      If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

4 In this question you should not use an approximating distribution.
At an election in Menham last year, \(24 \%\) of voters supported the Today Party. A student wishes to test whether support for the Today Party has decreased since last year. He chooses a random sample of 25 voters in Menham and finds that exactly 2 of them say that they support the Today Party. Test at the 5\% significance level whether support for the Today Party has decreased.

1 A six-sided die shows a six on 25 throws out of 200 throws. Test at the \(10 \%\) significance level the null hypothesis: P (throwing a six) \(= \frac { 1 } { 6 }\), against the alternative hypothesis: P (throwing a six) \(< \frac { 1 } { 6 }\).

3 When the council published a plan for a new road, only \(15 \%\) of local residents approved the plan. The council then published a revised plan and, out of a random sample of 300 local residents, 60 approved the revised plan. Is there evidence, at the \(2.5 \%\) significance level, that the proportion of local residents who approve the revised plan is greater than for the original plan?

2 A spinner has five sectors, each printed with a different colour. Susma and Sanjay both wish to test whether the spinner is biased so that it lands on red on fewer spins than it would if it were fair. Susma spins the spinner 40 times. She finds that it lands on red exactly 4 times.

1. Use a binomial distribution to carry out the test at the \(5 \%\) significance level.
   Sanjay also spins the spinner 40 times. He finds that it lands on red \(r\) times.
2. Use a binomial distribution to find the largest value of \(r\) that lies in the rejection region for the test at the 5\% significance level.

3 A researcher read a magazine article which stated that boys aged 1 to 3 prefer green to orange. It claimed that, when offered a green cube and an orange cube to play with, a boy is more likely to choose the green one. The researcher disagrees with this claim. She believes that boys of this age are equally likely to choose either colour. In order to test her belief, the researcher carried out a hypothesis test at the 5\% significance level. She offered a green cube and an orange cube to each of 10 randomly chosen boys aged 1 to 3 , and recorded the number, \(X\), of boys who chose the green cube. Out of the 10 boys, 8 boys chose the green cube.

1. 1. Assuming that the researcher's belief that either colour cube is equally likely to be chosen is valid, a student correctly calculates that \(\mathrm { P } ( X = 8 ) = 0.0439\), correct to 3 significant figures. He 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 on the researcher's claim that either colour cube is equally likely to be chosen.
2. Another researcher claims that a Type I error was made in carrying out the test. Explain why this cannot be true.
   A similar test, at the \(5 \%\) significance level, was carried out later using 10 other randomly chosen boys aged 1 to 3 .
3. Find the probability of a Type I error.

3 A factory owner models the number of employees who use the factory canteen on any day by the distribution \(\mathrm { B } ( 25 , p )\). In the past the value of \(p\) was 0.8 . A new menu is introduced in the canteen and the owner wants to test whether the value of \(p\) has increased. On a randomly chosen day he notes that the number of employees who use the canteen is 23 .

1. Use the binomial distribution to carry out the test at the \(10 \%\) significance level.
2. Given that there are 30 employees at the factory comment on the suitability of the owner's model.
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4 In a certain city it is necessary to pass a driving test in order to be allowed to drive a car. The probability of passing the driving test at the first attempt is 0.36 on average. A particular driving instructor claims that the probability of his pupils passing at the first attempt is higher than 0.36 . A random sample of 8 of his pupils showed that 7 passed at the first attempt.

1. Carry out an appropriate hypothesis test to test the driving instructor's claim, using a significance level of \(5 \%\).
2. In fact, most of this random sample happened to be careful and sensible drivers. State which type of error in the hypothesis test (Type I or Type II) could have been made in these circumstances and find the probability of this type of error when a sample of size 8 is used for the test.

1 At the 2009 election, \(\frac { 1 } { 3 }\) of the voters in Chington voted for the Citizens Party. One year later, a researcher questioned 20 randomly selected voters in Chington. Exactly 3 of these 20 voters said that if there were an election next week they would vote for the Citizens Party. Test at the \(2.5 \%\) significance level whether there is evidence of a decrease in support for the Citizens Party in Chington, since the 2009 election.

3 At an election in 2010, 15\% of voters in Bratfield voted for the Renewal Party. One year later, a researcher asked 30 randomly selected voters in Bratfield whether they would vote for the Renewal Party if there were an election next week. 2 of these 30 voters said that they would.

1. Use a binomial distribution to test, at the \(4 \%\) significance level, the null hypothesis that there has been no change in the support for the Renewal Party in Bratfield against the alternative hypothesis that there has been a decrease in support since the 2010 election.
2. (a) Explain why the conclusion in part (i) cannot involve a Type I error.
   (b) State the circumstances in which the conclusion in part (i) would involve a Type II error.

2 A hockey player found that she scored a goal on \(82 \%\) of her penalty shots. After attending a coaching course, she scored a goal on 19 out of 20 penalty shots. Making an assumption that should be stated, test at the 10\% significance level whether she has improved.

6 A machine is designed to generate random digits between 1 and 5 inclusive. Each digit is supposed to appear with the same probability as the others, but Max claims that the digit 5 is appearing less often than it should. In order to test this claim the manufacturer uses the machine to generate 25 digits and finds that exactly 1 of these digits is a 5 .

1. Carry out a test of Max's claim at the \(2.5 \%\) significance level.
2. Max carried out a similar hypothesis test by generating 1000 digits between 1 and 5 inclusive. The digit 5 appeared 180 times. Without carrying out the test, state the distribution that Max should use, including the values of any parameters.
3. State what is meant by a Type II error in this context.

2 Sami claims that he can read minds. He asks each of 50 people to choose one of the 5 letters A, B, C, D or E. He then tells each person which letter he believes they have chosen. He gets 13 correct. Sami says "This shows that I can read minds, because 13 is more than I would have got right if I were just guessing."

1. State null and alternative hypotheses for a test of Sami's claim.
2. Test at the \(10 \%\) significance level whether Sami's claim is justified.

7 A ten-sided spinner has edges numbered \(1,2,3,4,5,6,7,8,9,10\). Sanjeev claims that the spinner is biased so that it lands on the 10 more often than it would if it were unbiased. In an experiment, the spinner landed on the 10 in 3 out of 9 spins.

1. Test at the \(1 \%\) significance level whether Sanjeev's claim is justified.
2. Explain why a Type I error cannot have been made.
   In fact the spinner is biased so that the probability that it will land on the 10 on any spin is 0.5 .
3. Another test at the \(1 \%\) significance level, also based on 9 spins, is carried out. Calculate the probability of a Type II error.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

3 Sumitra has a six-sided die. She suspects that it is biased so that it shows a six less often than it would if it were fair. She decides to test the die by throwing it 30 times and noting the number of throws on which it shows a six.

1. It shows a six on exactly 2 throws. Use a binomial distribution to carry out the test at the \(5 \%\) significance level.
2. Later, Sumitra repeats the test at the \(5 \%\) significance level by throwing the die 30 times again. Find the probability of a Type I error in this second test.

8 The four sides of a spinner are \(A , B , C , D\). The spinner is supposed to be fair, but Sonam suspects that the spinner is biased so that the probability, \(p\), that it will land on side \(A\) is greater than \(\frac { 1 } { 4 }\). He spins the spinner 10 times and finds that it lands on side \(A 6\) times.

1. Test Sonam's suspicion using a \(1 \%\) significance level.
   Later Sonam carries out a similar test at the \(1 \%\) significance level, using another 10 spins of the spinner.
2. Calculate the probability of a Type I error.
3. Assuming that the value of \(p\) is actually \(\frac { 3 } { 5 }\), calculate the probability of a Type II error.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

2 Jill shoots arrows at a target. Last week, \(65 \%\) of her shots hit the target. This week Jill claims that she has improved. Out of her first 20 shots this week, she hits the target with 18 shots. Assuming shots are independent, test Jill's claim at the \(1 \%\) significance level.

6 A biscuit manufacturer claims that, on average, 1 in 3 packets of biscuits contain a prize offer. Gerry suspects that the proportion of packets containing the prize offer is less than 1 in 3 . In order to test the manufacturer's claim, he buys 20 randomly selected packets. He finds that exactly 2 of these packets contain the prize offer.

1. Carry out the test at the \(10 \%\) significance level.
2. Maria also suspects that the proportion of packets containing the prize offer is less than 1 in 3 . She also carries out a significance test at the \(10 \%\) level using 20 randomly selected packets. She will reject the manufacturer's claim if she finds that there are 3 or fewer packets containing the prize offer. Find the probability of a Type II error in Maria's test if the proportion of packets containing the prize offer is actually 1 in 7 .
3. Explain what is meant by a Type II error in this context.

6 In the past, Angus found that his train was late on \(15 \%\) of his daily journeys to work. Following a timetable change, Angus found that out of 60 randomly chosen days, his train was late on 6 days.

1. Test at the \(10 \%\) significance level whether Angus' train is late less often than it was before the timetable change.
   Angus used his random sample to find an \(\alpha \%\) confidence interval for the proportion of days on which his train is late. The upper limit of his interval was 0.150 , correct to 3 significant figures.
2. Calculate the value of \(\alpha\) correct to the nearest integer.

4 A train company claims that \(92 \%\) of trains on a particular line arrive on time. Sanjeep suspects that the true percentage is less than \(92 \%\). He chooses a random sample of 20 trains on this line and finds that exactly 16 of them arrive on time. Making an assumption that should be stated, test at the 5\% significance level whether Sanjeep's suspicion is justified.
[0pt] [6]

1 Isaac claims that \(30 \%\) of cars in his town are red. His friend Hardip thinks that the proportion is less than \(30 \%\). The boys decided to test Isaac's claim at the \(5 \%\) significance level and found that 2 cars out of a random sample of 18 were red. Carry out the hypothesis test and state your conclusion. [5]

6 It is claimed that a certain 6-sided die is biased so that it is more likely to show a six than if it was fair. In order to test this claim at the \(10 \%\) significance level, the die is thrown 10 times and the number of sixes is noted.

1. Given that the die shows a six on 3 of the 10 throws, carry out the test. On another occasion the same test is carried out again.
2. Find the probability of a Type I error.
3. Explain what is meant by a Type II error in this context.