Question

Tests and confidence intervals The $P-$value for a two-sided test of the null hypothesis $H_0:\mu =10$is$0.06$

a. Does the $95\%$ confidence interval for μ include $10$? Why or why not?

b. Does the $90\%$ confidence interval for μ include $10$? Why or why not?

Short answer

Part (a) Yes.

Part (b) No.

Step-by-step solution

Part (a) Step 1: Given information

$$\begin{gathered} H_0:\mu =10 \\ {H}_1:\mu \ne 10 \\ P=0.06 \end{gathered}$$

Part (a) Step 2: Explanation

If the $P-$value is lesser than the significance level then the null hypothesis is rejected.

$P=0.06>0.05=5\%\Rightarrow Failtoreject{H}_0$

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

The $95$ percent confidence interval comprises the value given in the null hypothesis $10$ because the null hypothesis failed to reject at the $5\%$ significance level.

Part (b) Step 1: Explanation

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

$P=0.06<0.10=10\%\Rightarrow Reject{H}_0$

A significance test at the $10\%$ significance level is associated with a $90$ percent confidence interval.

The $95$ percent confidence interval does not have the value specified in the null hypothesis $10$ since the null hypothesis was rejected at the $10\%$ significance level.