Binomial and Poisson tests: mixed practice Problem set A

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

1. (a) A hospital ward has an average of 3.2 admissions per night. The manager thinks that a new rota has reduced this.
2. (b) A beekeeper counts an average of 7.4 wasps per hive per day. She suspects that the number has increased.

A weaver finds an average of 0.35 flaws per metre of cloth. After a change of loom she checks 20 metres and finds 12 flaws.

1. (a) State the distribution of the number of flaws in 20 metres, if the mean has not changed.
2. (b) Test, at the 5% significance level, whether the mean number of flaws per metre has increased.

A stretch of coast has an average of 24 foggy mornings per year. This year there were 33. Is there evidence, at the 5% significance level, that the mean number of foggy mornings per year has increased?

A wildlife camera records an average of 1.7 foxes per night in a wood. A ranger will watch for the next 14 nights and test, at the 5% significance level, whether the mean number of foxes per night has increased.

1. (a) Find the critical region for the test.
2. (b) In the 14 nights there are 34 foxes. State the conclusion of the test.

A council says that 25% of residents cycle to work. A campaigner thinks that the true percentage is lower. She asks a random sample of 80 residents and tests her belief at the 5% significance level. What is the greatest percentage of the sample who could cycle to work for the council's statement to be rejected?

The number of breakdowns of a lift per day is modelled by a Poisson distribution with mean 1.5. After servicing, the engineer wants to test, at the 5% significance level, whether the mean number of breakdowns per day has decreased. The test uses the count on a single day for parts (a) and (b), and the count over 6 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 6 days.

A printer jams on average 0.8 times per day. After maintenance it jams 3 times in 14 days. The technician says that the mean number of jams per day has decreased.

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

A 1-in-60-years heatwave is taken to mean that the probability of such a heatwave in any year is 0.0167. In the last 15 years there have been 2 such heatwaves. A climate analyst claims that these heatwaves have become more frequent.

1. (a) Use the binomial model with p = 0.0167 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 school says that 20% of its students walk to school. A governor thinks that the true percentage is lower and tests this at the 5% significance level, using a random sample of students.

1. (a) Find the greatest percentage of a sample of 40 that gives sufficient evidence to reject the claim.
2. (b) Find the greatest percentage of a sample of 160 that gives sufficient evidence to reject the claim.
3. (c) Explain why the two percentages are different.

The number of cracks per day found in a bridge's concrete is modelled by a Poisson distribution with mean 0.8 per day. After repairs, an inspector will count the cracks over n days and test, at the 5% significance level, whether the mean number per day has decreased. Find the smallest value of n for which the test has a critical region.