One-tailed test (increase or decrease)

3 The local council claims that the average number of accidents per year on a particular road is 0.8 . Jane claims that the true average is greater than 0.8 . She looks at the records for a random sample of 3 recent years and finds that the total number of accidents during those 3 years was 5 .

1. Assume that the number of accidents per year follows a Poisson distribution.
   1. State null and alternative hypotheses for a test of Jane's claim.
   2. Test at the \(5 \%\) significance level whether Jane's claim is justified.
2. Jane finds that the number of accidents per year has been gradually increasing over recent years. State how this might affect the validity of the test carried out in part (a)(ii).

4 The number of cars arriving at a certain road junction on a weekday morning has a Poisson distribution with mean 4.6 per minute. Traffic lights are installed at the junction and council officer wishes to test at the \(2 \%\) significance level whether there are now fewer cars arriving. He notes the number of cars arriving during a randomly chosen 2 -minute period.

1. State suitable null and alternative hypotheses for the test.
2. Find the critical region for the test.
   The officer notes that, during a randomly chosen 2 -minute period on a weekday morning, exactly 5 cars arrive at the junction.
3. Carry out the test.
4. State, with a reason, whether it is possible that a Type I error has been made in carrying out the test in part (c).
   The number of cars arriving at another junction on a weekday morning also has a Poisson distribution with mean 4.6 per minute.
5. Use a suitable approximating distribution to find the probability that more than 300 cars will arrive at this junction in an hour.

7 The number of accidents per week at a certain factory has a Poisson distribution. In the past the mean has been 1.9 accidents per week. Last year, the manager gave all his employees a new booklet on safety. He decides to test, at the 5\% significance level, whether the mean number of accidents has been reduced. He notes the number of accidents during 4 randomly chosen weeks this year.

1. State suitable null and alternative hypotheses for the test.
2. Find the critical region for the test and state the probability of a Type I error.
3. State what is meant by a Type I error in this context.
4. During the 4 randomly chosen weeks there are a total of 3 accidents. State the conclusion that the manager should reach. Give a reason for your answer.
5. Assuming that the mean remains 1.9 accidents per week, use a suitable approximation to calculate the probability that there will be more than 100 accidents during a 52-week period.
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8 A new light was installed on a certain footpath. A town councillor decided to use a hypothesis test to investigate whether the number of people using the path in the evening had increased. Before the light was installed, the mean number of people using the path during any 20 -minute period during the evening was 1.01. After the light was installed, the total number, \(n\), of people using the path during 3 randomly chosen 20 -minute periods during the evening was noted.

1. Given that the value of \(n\) was 6 , use a Poisson distribution to carry out the test at the \(5 \%\) significance level.
2. Later a similar test, at the \(5 \%\) significance level, was carried out using another 3 randomly chosen 20 -minute periods during the evening. Find the probability of a Type I error.
3. State what is meant by a Type I error in this context.
4. State, in context, what further information would be needed in order to find the probability of a Type II error. Do not carry out any further calculation.
   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 The number of sports injuries per month at a certain college has a Poisson distribution. In the past the mean has been 1.1 injuries per month. The principal recently introduced new safety guidelines and she decides to test, at the \(2 \%\) significance level, whether the mean number of sports injuries has been reduced. She notes the number of sports injuries during a 6-month period.

1. Find the critical region for the test and state the probability of a Type I error.
2. State what is meant by a Type I error in this context.
3. During the 6 -month period there are a total of 2 sports injuries. Carry out the test.
4. Assuming that the mean remains 1.1 , calculate the probability that there will be fewer than 30 sports injuries during a 36-month period.

7 In the past the number of accidents per month on a certain road was modelled by a random variable with distribution \(\operatorname { Po } ( 0.47 )\). After the introduction of speed restrictions, the government wished to test, at the 5\% significance level, whether the mean number of accidents had decreased. They noted the number of accidents during the next 12 months. It is assumed that accidents occur randomly and that a Poisson model is still appropriate.

1. Given that the total number of accidents during the 12 months was 2 , carry out the test.
2. Explain what is meant by a Type II error in this context.
   It is given that the mean number of accidents per month is now in fact 0.05 .
3. Using another random sample of 12 months the same test is carried out again, with the same significance level. Find the probability of a Type II error.

6 The number of injuries per month at a certain factory has a Poisson distribution. In the past the mean was 2.1 injuries per month. New safety procedures are put in place and the management wishes to use the next 3 months to test, at the \(2 \%\) significance level, whether there are now fewer injuries than before, on average.

1. Find the critical region for the test.
2. Find the probability of a Type I error.
3. During the next 3 months there are a total of 3 injuries. Carry out the test.
4. Assuming that the mean remains 2.1 , calculate an estimate of the probability that there will be fewer than 20 injuries during the next 12 months.

5 The number of adult customers arriving in a shop during a 5-minute period is modelled by a random variable with distribution \(\operatorname { Po } ( 6 )\). The number of child customers arriving in the same shop during a 10 -minute period is modelled by an independent random variable with distribution \(\mathrm { Po } ( 4.5 )\).

1. Find the probability that during a randomly chosen 2 -minute period, the total number of adult and child customers who arrive in the shop is less than 3 .
2. During a sale, the manager claims that more adult customers are arriving than usual. In a randomly selected 30 -minute period during the sale, 49 adult customers arrive. Test the manager's claim at the 2.5\% significance level.

4 The number of faults in cloth made on a certain machine has a Poisson distribution with mean 2.4 per \(10 \mathrm {~m} ^ { 2 }\). An adjustment is made to the machine. It is required to test at the \(5 \%\) significance level whether the mean number of faults has decreased. A randomly selected \(30 \mathrm {~m} ^ { 2 }\) of cloth is checked and the number of faults is found.

1. State suitable null and alternative hypotheses for the test.
2. Find the probability of a Type I error.
   Exactly 3 faults are found in the randomly selected \(30 \mathrm {~m} ^ { 2 }\) of cloth.
3. Carry out the test at the \(5 \%\) significance level.
   Later a similar test was carried out at the \(5 \%\) significance level, using another randomly selected \(30 \mathrm {~m} ^ { 2 }\) of cloth.
4. Given that the number of faults actually has a Poisson distribution with mean 0.5 per \(10 \mathrm {~m} ^ { 2 }\), find the probability of a Type II error. \(5 X\) is a random variable with distribution \(\mathrm { B } ( 10,0.2 )\). A random sample of 160 values of \(X\) is taken.
5. Find the approximate distribution of the sample mean, including the values of the parameters.
6. Hence find the probability that the sample mean is less than 1.8 .

5 In the past the number of enquiries per minute at a customer service desk has been modelled by a random variable with distribution \(\operatorname { Po } ( 0.31 )\). Following a change in the position of the desk, it is expected that the mean number of enquiries per minute will increase. In order to test whether this is the case, the total number of enquiries during a randomly chosen 5-minute period is noted. You should assume that a Poisson model is still appropriate. Given that the total number of enquiries is 5 , carry out the test at the \(2.5 \%\) significance level.

5 In the past the number of enquiries per minute at a customer service desk has been modelled by a random variable with distribution \(\operatorname { Po } ( 0.31 )\). Following a change in the position of the desk, it is expected that the mean number of enquiries per minute will increase. In order to test whether this is the case, the total number of enquiries during a randomly chosen 5-minute period is noted. You should assume that a Poisson model is still appropriate. Given that the total number of enquiries is 5, carry out the test at the \(2.5 \%\) significance level.

7 The number of accidents per year on a certain road has the distribution \(\operatorname { Po } ( \lambda )\). In the past the value of \(\lambda\) was 3.3 . Recently, a new speed limit was imposed and the council wishes to test whether the value of \(\lambda\) has decreased. The council notes the total number, \(X\), of accidents during two randomly chosen years after the speed limit was introduced and it carries out a test at the \(5 \%\) significance level.

1. Calculate the probability of a Type I error.
2. Given that \(X = 2\), carry out the test. \includegraphics[max width=\textwidth, alt={}, center]{acd6f1c9-bbaf-40ca-b5cb-8466ddb9f596-10_2716_40_109_2010} \includegraphics[max width=\textwidth, alt={}, center]{acd6f1c9-bbaf-40ca-b5cb-8466ddb9f596-11_2716_29_107_22}
3. The council decides to carry out another similar test at the \(5 \%\) significance level using the same hypotheses and two different randomly chosen years. Given that the true value of \(\lambda\) is 0.6 , calculate the probability of a Type II error.
4. Using \(\lambda = 0.6\) and a suitable approximating distribution, find the probability that there will be more than 10 accidents in 30 years.
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7 The number of accidents per year on a certain road has the distribution \(\operatorname { Po } ( \lambda )\). In the past the value of \(\lambda\) was 3.3 . Recently, a new speed limit was imposed and the council wishes to test whether the value of \(\lambda\) has decreased. The council notes the total number, \(X\), of accidents during two randomly chosen years after the speed limit was introduced and it carries out a test at the \(5 \%\) significance level.

1. Calculate the probability of a Type I error.
2. Given that \(X = 2\), carry out the test. \includegraphics[max width=\textwidth, alt={}, center]{4f215475-30fd-47fb-aa77-0b53e339f50c-10_2718_35_107_2012} \includegraphics[max width=\textwidth, alt={}, center]{4f215475-30fd-47fb-aa77-0b53e339f50c-11_2716_29_107_22}
3. The council decides to carry out another similar test at the \(5 \%\) significance level using the same hypotheses and two different randomly chosen years. Given that the true value of \(\lambda\) is 0.6 , calculate the probability of a Type II error.
4. Using \(\lambda = 0.6\) and a suitable approximating distribution, find the probability that there will be more than 10 accidents in 30 years.
   If you use the following page to complete the answer to any question, the question number must be clearly shown.

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.

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 The number of new enquiries per day at an office has a Poisson distribution. In the past the mean has been 3 . Following a change of staff, the manager wishes to test, at the \(5 \%\) significance level, whether the mean has increased.

1. State the null and alternative hypotheses for this test. The manager notes the number, \(N\), of new enquiries during a certain 6 -day period. She finds that \(N = 25\) and then, assuming that the null hypothesis is true, she calculates that \(\mathrm { P } ( N \geqslant 25 ) = 0.0683\).
2. What conclusion should she draw?

6 The number of cases of asthma per month at a clinic has a Poisson distribution. In the past the mean has been 5.3 cases per month. A new treatment is introduced. In order to test at the \(5 \%\) significance level whether the mean has decreased, the number of cases in a randomly chosen month is noted.

1. Find the critical region for the test and, given that the number of cases is 2 , carry out the test.
2. Explain the meaning of a Type I error in this context and state the probability of a Type I error.
3. At another clinic the mean number of cases of asthma per month has the independent distribution \(\mathrm { Po } ( 13.1 )\). Assuming that the mean for the first clinic is still 5.3, use a suitable approximating distribution to estimate the probability that the total number of cases in the two clinics in a particular month is more than 20.

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.

4 The number of sightings of a golden eagle at a certain location has a Poisson distribution with mean 2.5 per week. Drilling for oil is started nearby. A naturalist wishes to test at the \(5 \%\) significance level whether there are fewer sightings since the drilling began. He notes that during the following 3 weeks there are 2 sightings.

1. Find the critical region for the test and carry out the test.
2. State the probability of a Type I error.
3. State why the naturalist could not have made a Type II error.

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.
   (a) State what is meant by a Type I error in this context.
   (b) State appropriate null and alternative hypotheses and find the probability of a Type I error.
   (c) 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 In the past the number of cars sold per day at a showroom has been modelled by a random variable with distribution \(\operatorname { Po } ( 0.7 )\). Following an advertising campaign, it is hoped that the mean number of sales per day will increase. In order to test at the \(10 \%\) significance level whether this is the case, the total number of sales during the first 5 days after the campaign is noted. You should assume that a Poisson model is still appropriate.

1. Given that the total number of cars sold during the 5 days is 5 , carry out the test.
   The number of cars sold per day at another showroom has the independent distribution \(\operatorname { Po } ( 0.6 )\). Assume that the distribution for the first showroom is still \(\operatorname { Po } ( 0.7 )\).
2. Find the probability that the total number of cars sold in the two showrooms during 3 days is exactly 2 .

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.

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 The number of accidents per month at a certain road junction has a Poisson distribution with mean 4.8. A new road sign is introduced warning drivers of the danger ahead, and in a subsequent month 2 accidents occurred.

1. A hypothesis test at the \(10 \%\) level is used to determine whether there were fewer accidents after the new road sign was introduced. Find the critical region for this test and carry out the test.
2. Find the probability of a Type I error. \(5 X\) and \(Y\) are independent random variables each having a Poisson distribution. \(X\) has mean 2.5 and \(Y\) has mean 3.1.
3. Find \(\mathrm { P } ( X + Y > 3 )\).
4. A random sample of 80 values of \(X\) is taken. Find the probability that the sample mean is less than 2.4.

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.