2.05b Hypothesis test for binomial proportion

1 In a game, a ball is thrown and lands in one of 4 slots, labelled \(A , B , C\) and \(D\). Raju wishes to test whether the probability that the ball will land in slot \(A\) is \(\frac { 1 } { 4 }\).

1. State suitable null and alternative hypotheses for Raju's test.
   The ball is thrown 100 times and it lands in slot \(A 15\) times.
2. Use a suitable approximating distribution to carry out the test at the \(2 \%\) significance level.

6 It is known that \(8 \%\) of adults in a certain town own a Chantor car. After an advertising campaign, a car dealer wishes to investigate whether this proportion has increased. He chooses a random sample of 25 adults from the town and notes how many of them own a Chantor car.

1. He finds that 4 of the 25 adults own a Chantor car. Carry out a hypothesis test at the 5\% significance level.
2. Explain which of the errors, Type I or Type II, might have been made in carrying out the test in part (a).
   Later, the car dealer takes another random sample of 25 adults from the town and carries out a similar hypothesis test at the 5\% significance level.
3. Find the probability of a Type I 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 The number of calls per day to an enquiry desk has a Poisson distribution. In the past the mean has been 5 . In order to test whether the mean has changed, the number of calls on a random sample of 10 days was recorded. The total number of calls was found to be 61 . Use an approximate distribution to test at the 10\% significance level whether the mean has changed.

2 Marie claims that she can predict the winning horse at the local races. There are 8 horses in each race. Nadine thinks that Marie is just guessing, so she proposes a test. She asks Marie to predict the winners of the next 10 races and, if she is correct in 3 or more races, Nadine will accept Marie's claim.

1. State suitable null and alternative hypotheses.
2. Calculate the probability of a Type I error.
3. State the significance level of the test.

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) \(< \frac { 1 } { 6 }\).

2 Jacques is a chef. He claims that \(90 \%\) of his customers are satisfied with his cooking. Marie suspects that the true percentage is lower than \(90 \%\). She asks a random sample of 15 of Jacques' customers whether they are satisfied. She then performs a hypothesis test of the null hypothesis \(p = 0.9\) against the alternative hypothesis \(p < 0.9\), where \(p\) is the population proportion of customers who are satisfied. She decides to reject the null hypothesis if fewer than 12 customers are satisfied.

1. In the context of the question, explain what is meant by a Type I error.
2. Find the probability of a Type I error in Marie's test.

4 It is claimed that 1 in every 4 packets of certain biscuits contains a free gift. Marisa and André both suspect that the true proportion is less than 1 in 4.

1. Marisa chooses 20 packets at random. She decides that if fewer than 3 contain free gifts, she will conclude that the claim is not justified. Use a binomial distribution to find the probability of a Type I error.
2. André chooses 25 packets at random. He decides to carry out a significance test at the \(1 \%\) level, using a binomial distribution. Given that only 1 of the 25 packets contains a free gift, carry out the test.

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?

5

1. The proportion of people having a particular medical condition is 1 in 100000 . A random sample of 2500 people is obtained. The number of people in the sample having the condition is denoted by \(X\).
   1. State, with a justification, a suitable approximating distribution for \(X\), giving the values of any parameters.
   2. Use the approximating distribution to calculate \(\mathrm { P } ( X > 0 )\).
2. The percentage of people having a different medical condition is thought to be \(30 \%\). A researcher suspects that the true percentage is less than \(30 \%\). In a medical trial a random sample of 28 people was selected and 4 people were found to have this condition. Use a binomial distribution to test the researcher's suspicion at the \(2 \%\) significance level.

6 A machine is supposed to produce random digits. Bob thinks that the machine is not fair and that the probability of it producing the digit 0 is less than \(\frac { 1 } { 10 }\). In order to test his suspicion he notes the number of times the digit 0 occurs in 30 digits produced by the machine. He carries out a test at the \(10 \%\) significance level.

1. State suitable null and alternative hypotheses.
2. Find the rejection region for the test.
3. State the probability of a Type I error.
   It is now given that the machine actually produces a 0 once in every 40 digits, on average.
4. Find the probability of a Type II error.
5. Explain the meaning of a Type II error in this context.

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.

6 At a certain hospital it was found that the probability that a patient did not arrive for an appointment was 0.2 . The hospital carries out some publicity in the hope that this probability will be reduced. They wish to test whether the publicity has worked. A random sample of 30 appointments is selected and the number of patients that do not arrive is noted. This figure is used to carry out a test at the \(5 \%\) significance level.

1. Explain why the test is one-tailed and state suitable null and alternative hypotheses.
2. Use a binomial distribution to find the critical region, and find the probability of a Type I error.
3. In fact 3 patients out of the 30 do not arrive. State the conclusion of the test, explaining your answer.

1 Each multiple choice question in a test has 4 suggested answers, exactly one of which is correct. Rehka knows nothing about the subject of the test, but claims that she has a special method for answering the questions that is better than just guessing. There are 60 questions in the test, and Rehka gets 22 correct.

1. State null and alternative hypotheses for a test of Rehka's claim.
2. Using a normal approximation, test at the \(5 \%\) significance level whether Rehka's claim is justified.

4 At a certain airport 20\% of people take longer than an hour to check in. A new computer system is installed, and it is claimed that this will reduce the time to check in. It is decided to accept the claim if, from a random sample of 22 people, the number taking longer than an hour to check in is either 0 or 1 .

1. Calculate the significance level of the test.
2. State the probability that a Type I error occurs.
3. Calculate the probability that a Type II error occurs if the probability that a person takes longer than an hour to check in is now 0.09 .

5 When a guitar is played regularly, a string breaks on average once every 15 months. Broken strings occur at random times and independently of each other.

1. Show that the mean number of broken strings in a 5 -year period is 4 . A guitar is fitted with a new type of string which, it is claimed, breaks less frequently. The number of broken strings of the new type was noted after a period of 5 years.
2. The mean number of broken strings of the new type in a 5 -year period is denoted by \(\lambda\). Find the rejection region for a test at the \(10 \%\) significance level when the null hypothesis \(\lambda = 4\) is tested against the alternative hypothesis \(\lambda < 4\).
3. Hence calculate the probability of making a Type I error. The number of broken guitar strings of the new type, in a 5 -year period, was in fact 1 .
4. State, with a reason, whether there is evidence at the \(10 \%\) significance level that guitar strings of the new type break less frequently.

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.

7 A hospital patient's white blood cell count has a Poisson distribution. Before undergoing treatment the patient had a mean white blood cell count of 5.2. After the treatment a random measurement of the patient's white blood cell count is made, and is used to test at the \(10 \%\) significance level whether the mean white blood cell count has decreased.

1. State what is meant by a Type I error in the context of the question, and find the probability that the test results in a Type I error.
2. Given that the measured value of the white blood cell count after the treatment is 2 , carry out the test.
3. Find the probability of a Type II error if the mean white blood cell count after the treatment is actually 4.1.

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.

4 At a power plant, the number of breakdowns per year has a Poisson distribution. In the past the mean number of breakdowns per year has been 4.8. Following some repairs, the management carry out a hypothesis test at the 5\% significance level to determine whether this mean has decreased. If there is at most 1 breakdown in the following year, they will conclude that the mean has decreased.

1. State what is meant by a Type I error in this context.
2. Find the probability of a Type I error.
3. Find the probability of a Type II error if the mean is now 0.9 breakdowns per year.

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.

2 Cloth made at a certain factory has been found to have an average of 0.1 faults per square metre. Suki claims that the cloth made by her machine contains, on average, more than 0.1 faults per square metre. In a random sample of \(5 \mathrm {~m} ^ { 2 }\) of cloth from Suki's machine, it was found that there were 2 faults. Assuming that the number of faults per square metre has a Poisson distribution,

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

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.