AS/A2 Stats 1 Stats 3

S2 Topic 4: Hypothesis tests
Testing the mean l of a Poisson distribution
You will be given a number of hypothesis tests. In each case a null and alternative hypothesis will be suggested, where l refers to the mean of the population Poisson distribution P(l), and a level of test. The results of a Poisson experiment will provided, giving the average number of occurrences in a given time frame. You must decide if the hypothesis test should have a non-significant or significant result based on the results of the Poisson experiement. Choose values for l and x, and use this form to obtain Poisson probabilities to help you complete the test below.

l: 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0 7.5 8.0 8.5 9.0 9.5 10.0 x: 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 P(X £ x) = P(X ³ x) =

Correct:

Attempts:

% Correct:

1	H0: l= 8 H1: l <8 Level of test: 5% Poisson experiment: 3 occurrences
	Non-significant	Significant
2	H0: l= 6.5 H1: l <6.5 Level of test: 1% Poisson experiment: 2 occurrences
	Non-significant	Significant
3	H0: l= 5.5 H1: l >5.5 Level of test: 5% Poisson experiment: 8 occurrences
	Non-significant	Significant
4	H0: l= 5.5 H1: l >5.5 Level of test: 20% Poisson experiment: 8 occurrences
	Non-significant	Significant
5	H0: l= 4.5 H1: l >4.5 Level of test: 10% Poisson experiment: 8 occurrences
	Non-significant	Significant
6	H0: l= 10 H1: l <10 Level of test: 5% Poisson experiment: 4 occurrences
	Non-significant	Significant
7	H0: l= 3 H1: l >3 Level of test: 5% Poisson experiment: 6 occurrences
	Non-significant	Significant
8	H0: l= 8 H1: l >8 Level of test: 5% Poisson experiment: 14 occurrences
	Non-significant	Significant
9	H0: l= 8 H1: l >8 Level of test: 1% Poisson experiment: 14 occurrences
	Non-significant	Significant
10	H0: l= 8 H1: l >8 Level of test: 1% Poisson experiment: 16 occurrences
	Non-significant	Significant

Summary In any hypothesis test we are testing the evidence to see if it is sufficient to reject the null hypothesis. A significant result implies: the null hypothesis is rejected the alternative hypothesis is accepted as true there is a chance that this conclusion is wrong and that the null hypothsis is in fact true. This is called a Type I error the probability of a Type I error is given by the level of the test A non-significant result implies: the null hypothesis is not rejected there is a chance that this conclusion is wrong and that the null hypothsis is in fact false. This is called a Type II error © MathsNet 2001