Question

The standardized test statistic for a test of $H_0:p=0.4$versus $H_a:pnotequalto0.4$is$z=2.43$This test is

a. not significant at either $\alpha =0.05$or $\alpha =0.01$

b. significant at $\alpha =0.05$but not at$\alpha =0.01$

c. significant at$\alpha =0.01$but not at $\alpha =0.05$

d. significant at both $\alpha =0.05$and$\alpha =0.01$

e. inconclusive because we don’t know the value of $\hat{p}$

Short answer

The test is significant at $\alpha =0.05$and $\alpha =0.01$

The correct option is$\left(b\right)$

Step-by-step solution

Given information

$z=2.43$

Level of significance $\left(\alpha \right)=0.05/0.01$

The null and alternative hypotheses are as follows:

$$\begin{gathered} H_0:p=0.4 \\ {H}_a:pisnotequalto0.4 \end{gathered}$$

The objective is to find the test statistics values at which test is significant 

The -value can be computed as:

$$\begin{gathered} p-value=2\times P(Z>\left|z\right|) \\ =2\times P(Z>\left|2.43\right|) \\ =2\times 0.0075 \\ =0.0150 \end{gathered}$$

If the null hypothesis is rejected, the test will be significant.

If the $p-value>\alpha$null hypothesis is rejected.

$p-value(0.0150)<\alpha (0.05)=$Reject the null hypothesis

$p-value(0.0150)>\alpha (0.01)=$Fail to reject the null hypothesis

At $\alpha =0.05$the test is considered significant.

Therefore, the correct option is$\left(b\right)$