Question

conduct a test at the \\( \alpha = 0.10 \\) level of significance by determining (a) the null and alternative hypotheses, (b) the test statistic, and (c) the p - value. assume the samples were obtained independently from a large population using simple random sampling. test whether \\( p _ { 1 } > p _ { 2 } \\). the sample data are \\( x _ { 1 } = 126, n _ { 1 } = 257, x _ { 2 } = 135 \\), and \\( n _ { 2 } = 317 \\). (a) choose the correct null and alternative hypotheses below. a. \\( h _ { 0 } : p _ { 1 } = p _ { 2 } \\) versus \\( h _ { 1 } : p _ { 1 } > p _ { 2 } \\) b. \\( h _ { 0 } : p _ { 1 } = p _ { 2 } \\) versus \\( h _ { 1 } : p _ { 1 } < p _ { 2 } \\) c. \\( h _ { 0 } : p _ { 1 } = 0 \\) versus \\( h _ { 1 } : p _ { 1 } > p _ { 2 } \\) d. \\( h _ { 0 } : p _ { 1 } = p _ { 2 } \\) versus \\( h _ { 1 } : p _ { 1 } \
eq p _ { 2 } \\) (b) determine the test statistic. \\( z _ { 0 } = \\) (round to two decimal places as needed.)

Explanation:

Step1: Calculate sample proportions

The sample proportion \(\hat{p}_1=\frac{x_1}{n_1}\), \(\hat{p}_2 = \frac{x_2}{n_2}\), and \(\hat{p}=\frac{x_1 + x_2}{n_1 + n_2}\).
\(\hat{p}_1=\frac{126}{257}\approx0.490\), \(\hat{p}_2=\frac{135}{317}\approx0.426\), \(\hat{p}=\frac{126 + 135}{257+317}=\frac{261}{574}\approx0.455\)

Step2: Calculate the test - statistic

The formula for the test - statistic \(z_0\) in a two - proportion z - test is \(z_0=\frac{\hat{p}_1-\hat{p}_2}{\sqrt{\hat{p}(1 - \hat{p})(\frac{1}{n_1}+\frac{1}{n_2})}}\)
Substitute the values:

$$
LATEXBLOCK0
$$

Answer:

\(z_0\approx1.53\)