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 α = 0.08, is there enough evidence to support the administrators claim? complete parts (a) through (e).

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} )

(e) interpret your decision in the context of the original claim.
at the 8% significance level, there enough evidence to the administrators claim that the mean score for the states eighth graders on the exam is more than 270.

Explanation:

Step1: Recall the decision rule

If \(P - value<\alpha\), we reject the null hypothesis \(H_0\). Here, \(\alpha = 0.08\) and \(P - value=0.043\).

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

Since \(0.043<0.08\) (i.e., \(P - value<\alpha\)).

Answer:

At the \(8\%\) significance level, there is enough evidence to support the administrator's claim that the mean score for the state's eighth graders on the exam is more than \(270\).