Question

use technology and a t - test to test the claim about the population mean \\( \mu \\) at the given level of significance \\( \alpha \\) using the given sample statistics. assume the population is normally distributed.
claim: \\( \mu>78 ; \alpha = 0.01 \\) sample statistics: \\( \overline { x } = 80.1, s = 3.4, n = 25 \\)
c. \\( \

$$\begin{array} { l } { h _ { 0 } : \\mu = 78 } \\\\ { h _ { a } : \\mu \ eq 78 } \\end{array}$$

\\) d. \\( \

$$\begin{array} { l } { h _ { 0 } : \\mu \\geq 78 } \\\\ { h _ { a } : \\mu < 78 } \\end{array}$$

\\)
what is the value of the standardized test statistic?
the standardized test statistic is 3.09. (round to two decimal places as needed.)
what is the p - value of the test statistic?
\\( p - value = 0.003 \\) (round to three decimal places as needed.)
decide whether to reject or fail to reject the null hypothesis. choose the correct answer below.
a. reject \\( h _ { 0 } \\). there is not enough evidence to support the claim.
b. reject \\( h _ { 0 } \\). there is enough evidence to support the claim.
c. fail to reject \\( h _ { 0 } \\). there is enough evidence to support the claim.
d. fail to reject \\( h _ { 0 } \\). there is not enough evidence to support the claim.

Explanation:

Step1: Recall the decision rule for t - test

If \(P - value\leq\alpha\), reject \(H_0\). Here, \(\alpha = 0.01\) and \(P - value=0.003\).

Step2: Compare \(P - value\) and \(\alpha\)

Since \(0.003<0.01\) (i.e., \(P - value\leq\alpha\)).

Answer:

B. Reject \(H_0\). There is enough evidence to support the claim.