AS/A2 Stats 1 Stats 2

S3 Topic 3: Estimation, confidence intervals and tests
Testing the paramater p of a Binomial distribution
You will be given a number of hypothesis tests. In each case a null and alternative hypothesis will be suggested, where p refers to the probability of the population Binomial distribution B(n,p), and a level of test. The results of a Binomial experiment will provided, giving the number of successes in a number of trials. You will need to use the Normal distribution, N(m,s2) as an approximation to the Binomial, where
m = np, s2 = np(1-p)
You must decide if the hypothesis test should have a non-significant or significant result based on the results of the Binomial experiement. A table of critical values is provided but you will have to work out the test statistic yourself, using:
where the mean is taken to be the number of observed successes

Critical values	One-tailed	One-tailed	Two-tailed
10%	1.282	-1.282	±1.645
5%	1.645	-1.645	±1.96
2.5%	1.96	-1.96	±2.241
1%	2.326	-2.326	±2.575

Correct:

Attempts:

% Correct:

1	H0: p= 0.5 H1: p ¹0.5 Level of test: 5% Binomial experiment: 41 successes out of 100
	Non-significant	Significant
2	H0: p= 0.5 H1: p <0.5 Level of test: 5% Binomial experiment: 41 successes out of 100
	Non-significant	Significant
3	H0: p= 0.5 H1: p ¹0.5 Level of test: 5% Binomial experiment: 25 successes out of 40
	Non-significant	Significant
4	H0: p= 0.5 H1: p >0.5 Level of test: 2.5% Binomial experiment: 60 successes out of 100
	Non-significant	Significant
5	H0: p= 0.6 H1: p >0.6 Level of test: 5% Binomial experiment: 100 successes out of 150
	Non-significant	Significant
6	H0: p= 0.6 H1: p ¹0.6 Level of test: 5% Binomial experiment: 100 successes out of 150
	Non-significant	Significant
7	H0: p= 0.4 H1: p<0.4 Level of test: 10% Binomial experiment: 52 successes out of 150
	Non-significant	Significant
8	H0: p= 0.4 H1: p<0.4 Level of test: 5% Binomial experiment: 52 successes out of 150
	Non-significant	Significant
9	H0: p= 0.4 H1: p<0.4 Level of test: 1% Binomial experiment: 42 successes out of 150
	Non-significant	Significant
10	H0: p= 0.4 H1: p>0.4 Level of test: 5% Binomial experiment: 70 successes out of 150
	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