Question

a. test the claim using a hypothesis test
consider the first sample to be the sample of people over the age of 55 and the second sample to be the sample of people under the age of 25. what are the null and alternative hypotheses for the hypothesis test?
a. ( h_0: p_1
eq p_2 )
( h_1: p_1 = p_2 )
b. ( h_0: p_1 = p_2 )
( h_1: p_1
eq p_2 )
c. ( h_0: p_1 leq p_2 )
( h_1: p_1
eq p_2 )
d. ( h_0: p_1 geq p_2 )
( h_1: p_1
eq p_2 )
e. ( h_0: p_1 = p_2 )
( h_1: p_1 > p_2 )
f. ( h_0: p_1 = p_2 )
( h_1: p_1 < p_2 )
identify the test statistic
( z = square )
(round to two decimal places as needed )

Explanation:

To determine the null and alternative hypotheses for a hypothesis test comparing two proportions (\(p_1\) for people over 55 and \(p_2\) for people under 25), we analyze the options:

• The null hypothesis (\(H_0\)) typically states no difference or a specific relationship (equality here, \(p_1 = p_2\) for a two - proportion test when testing for a difference or a directional claim).
• The alternative hypothesis (\(H_1\)) reflects the claim. Option E has \(H_0:p_1 = p_2\) (null hypothesis assumes equality) and \(H_1:p_1>p_2\) (alternative hypothesis claims \(p_1\) is greater than \(p_2\)), which is a valid setup for testing if the proportion for the first group (over 55) is greater than the second (under 25). Other options have incorrect null - alternative pairings (e.g., A has \(H_0\) as inequality, which is not standard; C has an incorrect \(H_0\) direction, etc.).

For the test statistic \(z\), we use the formula for the z - test for two proportions:

$$z=\frac{(\hat{p}_1-\hat{p}_2)-0}{\sqrt{\bar{p}(1 - \bar{p})(\frac{1}{n_1}+\frac{1}{n_2})}}$$

where \(\hat{p}_1=\frac{x_1}{n_1}\), \(\hat{p}_2=\frac{x_2}{n_2}\), and \(\bar{p}=\frac{x_1 + x_2}{n_1 + n_2}\) (assuming we have sample counts \(x_1,x_2\) and sample sizes \(n_1,n_2\)). However, since the problem does not provide sample data (counts or proportions for each group), we can't calculate the numerical value of \(z\) here. But based on the hypothesis selection, the correct null and alternative hypotheses are from option E.

Answer:

E. \(H_0:p_1 = p_2\), \(H_1:p_1>p_2\) (For the test statistic \(z\), more information about sample proportions and sizes is needed to calculate its value.)