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 and 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?

\\( \bigcirc \\) a. \\( h _ { 0 } : \mu _ { 1 } = \mu _ { 2 } \\)
\\( h _ { a } : \mu _ { 1 } \
eq \mu _ { 2 } \\)
\\( \bigcirc \\) b. \\( h _ { 0 } : \mu _ { 1 } \
eq \mu _ { 2 } \\)
\\( h _ { a } : \mu _ { 1 } = \mu _ { 2 } \\)
\\( \bigcirc \\) c. \\( h _ { 0 } : \mu _ { 1 } = \mu _ { 2 } \\)
\\( h _ { a } : \mu _ { 1 } < \mu _ { 2 } \\)
\\( \bigcirc \\) d. \\( h _ { 0 } : \mu _ { 1 } = \mu _ { 2 } \\)
\\( h _ { a } : \mu _ { 1 } > \mu _ { 2 } \\)
\\( \bigcirc \\) e. \\( h _ { 0 } : \mu _ { 1 } > \mu _ { 2 } \\)
\\( h _ { a } : \mu _ { 1 } = \mu _ { 2 } \\)
\\( \bigcirc \\) f. \\( h _ { 0 } : \mu _ { 1 } < \mu _ { 2 } \\)
\\( h _ { a } : \mu _ { 1 } = \mu _ { 2 } \\)

next, compute the test statistic.

\\( t = - 3.69 \\) (round to three decimal places as needed.)

now determine the critical values.

\\( \pm t _ { \alpha / 2 } = \pm \square \\) (round to three decimal places as needed.)

Explanation:

Step1: Calculate the degrees of freedom

The degrees of freedom for the pooled \(t -\)test is \(df=n_1 + n_2-2\). Given \(n_1 = 14\) and \(n_2=14\), then \(df=14 + 14-2=26\).

Step2: Find the critical value

For a two - tailed test with \(\alpha = 0.05\), \(\frac{\alpha}{2}=0.025\). Using the \(t -\)distribution table or a calculator, the critical value \(t_{\alpha/2}\) with \(df = 26\) is \(t_{0.025,26}\).
From the \(t -\)distribution table or using a calculator (e.g., in R: qt(0.975,26)), we get \(t_{0.025,26}=2.056\).

Answer:

\(\pm2.056\)