Question

provided below are summary statistics for independent simple random samples from two populations. use the pooled t - test and the pooled t - interval procedure to conduct the required hypothesis test an obtain the specified confidence interval.
\\( \overline { x } _ { 1 } = 13, s _ { 1 } = 2.2, n _ { 1 } = 14, \overline { x } _ { 2 } = 16, s _ { 2 } = 2.1, n _ { 2 } = 14 \\)
a. two - tailed test, \\( \alpha = 0.05 \\)
b. 95% confidence interval
a. first, what are the correct hypotheses for a two - tailed test?
a \\( h _ { 0 } : \mu _ { 1 } = \mu _ { 2 } \\)
\\( h _ { a } : \mu _ { 1 } \
eq \mu _ { 2 } \\)
b. \\( h _ { 0 } : \mu _ { 1 } \
eq \mu _ { 2 } \\)
\\( h _ { a } : \mu _ { 1 } = \mu _ { 2 } \\)
c. \\( h _ { 0 } : \mu _ { 1 } = \mu _ { 2 } \\)
\\( h _ { a } : \mu _ { 1 } < \mu _ { 2 } \\)
d. \\( h _ { 0 } : \mu _ { 1 } = \mu _ { 2 } \\)
\\( h _ { a } : \mu _ { 1 } > \mu _ { 2 } \\)
e \\( h _ { 0 } : \mu _ { 1 } > \mu _ { 2 } \\)
\\( h _ { a } : \mu _ { 1 } = \mu _ { 2 } \\)
f. \\( h _ { 0 } : \mu _ { 1 } \leq \mu _ { 2 } \\)
\\( h _ { a } : \mu _ { 1 } = \mu _ { 2 } \\)
next, compute the test statistic.
\\( t = - 3.59 \\) (round to three decimal places as needed.)
now determine the critical values.
\\( \pm t _ { \alpha / 2 } = \pm 2.056 \\) (round to three decimal places as needed.)
what is the conclusion of the hypothesis test?
since the test statistic in the rejection region, \\( h _ { 0 } \\).

Explanation:

Step1: Determine the rejection region

For a two - tailed test with \(\alpha = 0.05\), the critical values are \(\pm t_{\alpha/2}=\pm2.056\). The rejection region is \(t < - 2.056\) or \(t>2.056\).

Step2: Compare the test statistic with the rejection region

The test statistic \(t=-3.59\). Since \(-3.59<-2.056\), the test statistic is in the rejection region.

Answer:

Since the test statistic is in the rejection region, reject \(H_0\).