2.05c Significance levels: one-tail and two-tail

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.

7 The number of absences by girls from a certain class on any day is modelled by a random variable with distribution \(\operatorname { Po } ( 0.2 )\). The number of absences by boys from the same class on any day is modelled by an independent random variable with distribution \(\operatorname { Po } ( 0.3 )\).

1. Find the probability that, during a randomly chosen 2-day period, the total number of absences is less than 3 .
2. Find the probability that, during a randomly chosen 5-day period, the number of absences by boys is more than 3.
3. The teacher claims that, during the football season, there are more absences by boys than usual. In order to test this claim at the 5\% significance level, he notes the number of absences by boys during a randomly chosen 5-day period during the football season.
   1. State what is meant by a Type I error in this context.
   2. State appropriate null and alternative hypotheses and find the probability of a Type I error.
   3. In fact there were 4 absences by boys during this period. Test the teacher's claim at the 5\% significance level.
      If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

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.

5 The manufacturer of a certain type of biscuit claims that \(10 \%\) of packets include a free offer printed on the packet. Jyothi suspects that the true proportion is less than \(10 \%\). He plans to test the claim by looking at 40 randomly selected packets and, if the number which include the offer is less than 2 , he will reject the manufacturer's claim.

1. State suitable hypotheses for the test.
2. Find the probability of a Type I error.
   On another occasion Jyothi looks at 80 randomly selected packets and finds that exactly 6 include the free offer.
3. Calculate an approximate \(90 \%\) confidence interval for the proportion of packets that include the offer.
4. Use your confidence interval to comment on the manufacturer's claim. \(6 X\) is a random variable with probability density function given by $$f ( x ) = \begin{cases} \frac { a } { x ^ { 2 } } & 1 \leqslant x \leqslant b \\ 0 & \text { otherwise } \end{cases}$$ where \(a\) and \(b\) are constants.

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.

2 Karim has noted the lifespans, in weeks, of a large random sample of certain insects. He carries out a test, at the \(1 \%\) significance level, for the population mean, \(\mu\). Karim's null hypothesis is \(\mu = 6.4\).

1. Given that Karim's test is two-tail, state the alternative hypothesis.
   Karim finds that the value of the test statistic is \(z = 2.43\).
2. Explain what conclusion he should draw.
3. Explain briefly when a one-tail test is appropriate, rather than a two-tail test.

5 A teacher models the numbers of girls and boys who arrive late for her class on any day by the independent random variables \(G \sim \operatorname { Po } ( 0.10 )\) and \(B \sim \operatorname { Po } ( 0.15 )\) respectively.

1. Find the probability that during a randomly chosen 2-day period no girls arrive late.
2. Find the probability that during a randomly chosen 5-day period the total number of students who arrive late is less than 3 .
3. It is given that the values of \(\mathrm { P } ( G = r )\) and \(\mathrm { P } ( B = r )\) for \(r \geqslant 3\) are very small and can be ignored. Find the probability that on a randomly chosen day more girls arrive late than boys.
   Following a timetable change the teacher claims that on average more students arrive late than before the change. During a randomly chosen 5-day period a total of 4 students are late.
4. Test the teacher's claim at the \(5 \%\) significance level.

6 The time, in minutes, for Anjan's journey to work on Mondays has mean 38.4 and standard deviation 6.9.

1. Find the probability that Anjan's mean journey time for a random sample of 30 Mondays is between 38 and 40 minutes.
   Anjan wishes to test whether his mean journey time is different on Tuesdays. He chooses a random sample of 30 Tuesdays and finds that his mean journey time for these 30 Tuesdays is 40.2 minutes. Assume that the standard deviation for his journey time on Tuesdays is 6.9 minutes.
2. 1. State, with a reason, whether Anjan should use a one-tail or a two-tail test.
   2. Carry out the test at the \(10 \%\) significance level.
   3. Explain whether it was necessary to use the Central Limit theorem in part (b)(ii).
      If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

6 In a certain factory the number of items per day found to be defective has had the distribution \(\operatorname { Po } ( 1.03 )\). After the introduction of new quality controls, the management wished to test at the \(10 \%\) significance level whether the mean number of defective items had decreased. They noted the total number of defective items produced in 5 randomly chosen days. It is assumed that defective items occur randomly and that a Poisson model is still appropriate.

1. Given that the total number of defective items produced during the 5 days was 2 , carry out the test.
2. Using another random sample of 5 days the same test is carried out again, with the same significance level. Find the probability of a Type I error.
3. Explain what is meant by a Type I error in this context.

2 A headteacher models the number of children who bring a 'healthy' packed lunch to school on any day by the distribution \(\mathrm { B } ( 150 , p )\). In the past, she has found that \(p = \frac { 1 } { 3 }\). Following the opening of a fast food outlet near the school, she wishes to test, at the \(1 \%\) significance level, whether the value of \(p\) has decreased.

1. State the null and alternative hypotheses for this test.
   On a randomly chosen day she notes the number, \(N\), of children who bring a 'healthy' packed lunch to school. She finds that \(N = 36\) and then, assuming that the null hypothesis is true, she calculates that \(\mathrm { P } ( N \leqslant 36 ) = 0.0084\).
2. State, with a reason, the conclusion that the headteacher should draw from the test.
3. According to the model, what is the largest number of children who might bring a packed lunch to school?

6 The number of accidents per month, \(X\), at a factory has a Poisson distribution. In the past the mean has been 1.1 accidents per month. Some new machinery is introduced and the management wish to test whether the mean has increased. They note the number of accidents in a randomly chosen month and carry out a hypothesis test at the 1\% significance level.

1. Show that the critical region for the test is \(X \geqslant 5\). Given that the number of accidents is 6 , carry out the test.
   Later they carry out a similar test, also at the \(1 \%\) significance level.
2. Explain the meaning of a Type I error in this context and state the probability of a Type I error.
3. Given that the mean is now 7.0 , find 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.

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.
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7 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.

1. It is suggested that the first 30 appointments on a Monday should be used for the test. Give a reason why this is not an appropriate sample.
   A suitable 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.
2. Explain why the test is one-tail and state suitable null and alternative hypotheses.
3. State what is meant by a Type I error in this context.
4. Use the binomial distribution to find the critical region, and find the probability of a Type I error.
5. In fact 3 patients out of the 30 do not arrive. State the conclusion of the test, explaining your answer.

1 The time taken for Samuel to drive home from work is distributed with mean 46 minutes. Samuel discovers a different route and decides to test at the \(5 \%\) level whether the mean time has changed. He tries this route on a large number of different days chosen randomly and calculates the mean time.

1. State the null and alternative hypotheses for this test.
2. Samuel calculates the value of his test statistic \(z\) to be - 1.729 . What conclusion can he draw?

2 In summer the growth rate of grass in a lawn has a normal distribution with mean 3.2 cm per week and standard deviation 1.4 cm per week. A new type of grass is introduced which the manufacturer claims has a slower growth rate. A hypothesis test of this claim at the \(5 \%\) significance level was carried out using a random sample of 10 lawns that had the new grass. It may be assumed that the growth rate of the new grass has a normal distribution with standard deviation 1.4 cm per week.

1. Find the rejection region for the test.
2. The probability of making a Type II error when the actual value of the mean growth rate of the new grass is \(m \mathrm {~cm}\) per week is less than 0.5 . Use your answer to part (i) to write down an inequality for \(m\).

4 It is not known whether a certain coin is fair or biased. In order to perform a hypothesis test, Raj tosses the coin 10 times and counts the number of heads obtained. The probability of obtaining a head on any throw is denoted by \(p\).

1. The null hypothesis is \(p = 0.5\). Find the acceptance region for the test, given that the probability of a Type I error is to be at most 0.1 .
2. Calculate the probability of a Type II error in this test if the actual value of \(p\) is 0.7 .

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.

7 In the past, the number of house sales completed per week by a building company has been modelled by a random variable which has the distribution \(\mathrm { Po } ( 0.8 )\). Following a publicity campaign, the builders hope that the mean number of sales per week will increase. In order to test at the \(5 \%\) significance level whether this is the case, the total number of sales during the first 3 weeks after the campaign is noted. It is assumed that a Poisson model is still appropriate.

1. Given that the total number of sales during the 3 weeks is 5 , carry out the test.
2. During the following 3 weeks the same test is carried out again, using the same significance level. Find the probability of a Type I error.
3. Explain what is meant by a Type I error in this context.
4. State what further information would be required in order to find the probability of a Type II error.

2 An engineering test consists of 100 multiple-choice questions. Each question has 5 suggested answers, only one of which is correct. Ashok knows nothing about engineering, but he claims that his general knowledge enables him to get more questions correct than just by guessing. Ashok actually gets 27 answers correct. Use a suitable approximating distribution to test at the \(5 \%\) significance level whether his claim is justified.

5 The management of a factory thinks that the mean time required to complete a particular task is 22 minutes. The times, in minutes, taken by employees to complete this task have a normal distribution with mean \(\mu\) and standard deviation 3.5. An employee claims that 22 minutes is not long enough for the task. In order to investigate this claim, the times for a random sample of 12 employees are used to test the null hypothesis \(\mu = 22\) against the alternative hypothesis \(\mu > 22\) at the \(5 \%\) significance level.

1. Show that the null hypothesis is rejected in favour of the alternative hypothesis if \(\bar { x } > 23.7\) (correct to 3 significant figures), where \(\bar { x }\) is the sample mean.
2. Find the probability of a Type II error given that the actual mean time is 25.8 minutes.

2 The heights of a certain type of plant have a normal distribution. When the plants are grown without fertilizer, the population mean and standard deviation are 24.0 cm and 4.8 cm respectively. A gardener wishes to test, at the \(2 \%\) significance level, whether Hiergro fertilizer will increase the mean height. He treats 150 randomly chosen plants with Hiergro and finds that their mean height is 25.0 cm . Assuming that the standard deviation of the heights of plants treated with Hiergro is still 4.8 cm , carry out the test.