Question

the test statistic of ( z = 1.26 ) is obtained when testing the claim that ( p
eq0.285 ).
a. identify the hypothesis test as being two - tailed, left - tailed, or right - tailed.
b. find the p - value.
c. using a significance level of ( alpha = 0.10 ), should we reject ( h_0 ) or should we fail to reject ( h_0 )?

Explanation:

Step1: Identify the type of test

The claim is \(p
eq0.285\). In hypothesis testing, when the alternative hypothesis is \(H_1:p
eq p_0\) (where \(p_0 = 0.285\) here), it is a two - tailed test.

Step2: Find the P - value

For a two - tailed test with \(z = 1.26\), we first find the area to the right of \(z = 1.26\) using the standard normal distribution table. The area to the left of \(z=1.26\) is \(P(Z < 1.26)=0.8962\) (from the standard normal table). The area to the right of \(z = 1.26\) is \(P(Z>1.26)=1 - 0.8962=0.1038\). Since it is a two - tailed test, \(P - value=2\times0.1038 = 0.2076\).

Step3: Make a decision

We are given \(\alpha=0.10\). The decision rule for a hypothesis test is: if \(P - value<\alpha\), reject \(H_0\); if \(P - value\geq\alpha\), fail to reject \(H_0\). Here, \(P - value = 0.2076\) and \(\alpha=0.10\). Since \(0.2076>0.10\), we fail to reject \(H_0\).

Answer:

a. This is a two - tailed test.
b. The \(P - value\) is \(0.2076\).
c. We should fail to reject \(H_0\).