Binomial and Poisson tests: mixed practice Problem set B

Write down H0 and H1 for each test. Define the parameter you use.

1. (a) A call centre receives an average of 8.5 calls per hour. The manager believes that it has become quieter since a new website opened.
2. (b) A junction has had an average of 1.3 accidents per week. Residents think that there are more since the speed limit was raised.

A web server logs an average of 0.45 errors per minute. After an update the server is watched for 15 minutes and logs 11 errors.

1. (a) State the distribution of the number of errors in 15 minutes, if the mean has not changed.
2. (b) Test, at the 5% significance level, whether the mean number of errors per minute has increased.

A mountain pass has an average of 41 days per year with snow on the road. Last year there were 29. Is there evidence, at the 5% significance level, that the mean number of snowy days per year has decreased?

A weather station records an average of 0.9 thunderstorms per week in summer. A meteorologist will look at the next 20 weeks and test, at the 5% significance level, whether the mean number of thunderstorms per week has increased.

1. (a) Find the critical region for the test.
2. (b) In the 20 weeks there are 25 thunderstorms. State the conclusion of the test.

A shop says that 40% of its customers pay by card. The owner thinks that the true percentage is lower. He asks a random sample of 60 customers and tests his belief at the 5% significance level. What is the greatest percentage of the sample who could pay by card for the shop's statement to be rejected?

The number of faults found in a batch of circuit boards per day is modelled by a Poisson distribution with mean 2.1. After a change of supplier, the manager wants to test, at the 5% significance level, whether the mean number of faults per day has decreased. The test uses the count on a single day for parts (a) and (b), and the count over 5 days for part (c).

1. (a) Show that there is no critical region.
2. (b) Find the smallest whole-number percentage level at which the test has a critical region.
3. (c) Find the critical region at the 5% level for the count over 5 days.

A gym receives an average of 2.4 complaints per week. After new equipment is installed, it receives 10 complaints in 8 weeks. The manager says that the mean number of complaints per week has decreased.

1. (a) Write down suitable hypotheses, defining the parameter.
2. (b) Test the claim at the 5% significance level.

A 1-in-30-years drought is taken to mean that the probability of a drought in any year is 0.035. In the last 12 years there have been 2 droughts. A farmer claims that droughts have become more frequent.

1. (a) Use the binomial model with p = 0.035 to test the claim at the 5% significance level.
2. (b) Use a Poisson model with the mean for the whole period to test the claim at the 5% significance level.
3. (c) Find the range of significance levels for which the two models give different conclusions.

A bakery says that 35% of its customers buy a loaf. The owner thinks that the true percentage is lower and tests this at the 5% significance level, using a random sample of customers.

1. (a) Find the greatest percentage of a sample of 50 that gives sufficient evidence to reject the claim.
2. (b) Find the greatest percentage of a sample of 200 that gives sufficient evidence to reject the claim.
3. (c) Explain why the two percentages are different.

The number of typing errors per page in a report is modelled by a Poisson distribution with mean 0.45 per page. After a training course, an editor will count the errors over n pages and test, at the 5% significance level, whether the mean number per page has decreased. Find the smallest value of n for which the test has a critical region.