Question

a random sample of 78 eighth grade students scores on a national mathematics assessment test has a mean score of 277. this test result prompts a state school administrator to declare that the mean score for the states eighth graders on this exam is more than 270. assume that the population standard deviation is 36. at \\( \alpha = 0.08 \\), is there enough evidence to support the administrators claim? complete parts (a) through (e).

a. \\( h _ { 0 } : mu leq 270 \\) (claim) \\( h _ { a } : mu > 270 \\)
b. \\( h _ { 0 } : mu geq 270 \\) (claim) \\( h _ { a } : mu < 270 \\)
c. \\( h _ { 0 } : mu leq 270 \\) \\( h _ { a } : mu > 270 \\) (claim)
d. \\( h _ { 0 } : mu = 270 \\) \\( h _ { a } : mu > 270 \\) (claim)
e. \\( h _ { 0 } : mu = 270 \\) (claim) \\( h _ { a } : mu > 270 \\)
f. \\( h _ { 0 } : mu < 270 \\) \\( h _ { a } : mu geq 270 \\) (claim)

(b) find the standardized test statistic z.
\\( z = 1.72 \\) (round to two decimal places as needed.)

(c) find the p - value.
\\( p - value = 0.043 \\) (round to three decimal places as needed.)

(d) decide whether to reject or fail to reject the null hypothesis.
reject \\( h _ { 0 } \\)
fail to reject \\( h _ { 0 } \\)

Explanation:

Step1: Compare P - value and significance level

We are given a significance level \(\alpha = 0.08\) and a P - value \(P=0.043\).

Step2: Make a decision

Since \(P = 0.043<\alpha=0.08\), we reject the null hypothesis.

Answer:

Reject \(H_{0}\)