Question

After checking that conditions are met, you perform a significance test of $H_0:\mu =1$versus $H_a:\mu \ne 1$You obtain a $P-$value of $0.022$Which of the following must be true?

a. A $95\%$confidence interval for ${\mu }^{\mu }$ will include the value $1$

b. A $95\%$confidence interval for ${\mu }^{\mu }$ will include the value $0.022$

c. A $99\%$confidence interval for ${\mu }^{\mu }$ will include the value$1$

d. A $99\%$confidence interval for ${\mu }^{\mu }$ will include the value $0.022$

e. None of these is necessarily true.

Short answer

The correct option is (c) A $99\%$ confidence interval for ${\mu }^{\mu }$ will include the value $1$

Step-by-step solution

Given information

$$\begin{gathered} H_0:\mu =1 \\ {H}_a:\mu \ne 1 \\ P=0.022 \end{gathered}$$

Calculation

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

$$\begin{gathered} P<0.05=5\%reject{H}_0 \\ P>0.01=1\%=failstoreject{H}_0 \end{gathered}$$

A significance test at the $5\%$ significance level equates to a $95$ percent confidence interval.

A significance test at the $1\%$ significance level equates to a $99$ percent confidence interval.

There is no one in $95\%$ of the confidence interval.

There are $1$ in the $95\%$ confidence interval.

So correct option is (c).