Question

Tests and confidence intervals The P-value for a two-sided test of the null hypothesis $H_0:\mu =15is0.03$

a. Does the $99\%$ confidence interval for $\mu$ include $15$? Why or why not?

b. Does the $95\%$ confidence interval for $\mu$ include $15$? Why or why not?

Short answer

Part (a) Yes.

Part (b) No.

Step-by-step solution

Part (a) Step 1: Given information

$$\begin{gathered} H_0:\mu =15 \\ {H}_1:\mu \ne 15 \\ P=0.03 \end{gathered}$$

Part (a) Step 2: Explanation

A significance test at the $100\%-99\%=1\%$significance level corresponds to a $99$percent confidence interval.

The null hypothesis is rejected if the P-value is less than the significance level.$P=0.03>0.01=1\%\Rightarrow Failtoreject{H}_0$

Because the null hypothesis failed to reject at the $1\%$ significance level, the null hypothesis value is used in the $99$ percent confidence interval.

Part (b) Step 1: Explanation

A significance test at the $100\%-95\%=5\%$significance level is associated with a $95$percent confidence interval.

The null hypothesis is rejected if the $P-$value is less than the significance level.

$P=0.03<0.05=5\%\Rightarrow Reject{H}_0$

The $95$ percent confidence interval does not have the value specified in the null hypothesis since the null hypothesis fails at the $5\%$ significance level.