Question

Consider the test of $H_0:\mu =7$. For each of the following, find the p-value of the test:

a.$H_a:\mu >7 ; z=1.20$

b.$H_a:\mu <7 ; z=-1.20$

c.$H_a:\mu \ne 7 ; z=1.20$

Short answer

1. The p-value is 0.115.
2. The p-value is 0.884.
3. The p-value is 0.230.

Step-by-step solution

 Given Information

The null hypothesis and the alternative hypothesis are given. Z- score is given.

From this, we can easily compute the p-value.

(a) Calculate

The hypothesis are given by

$$\begin{gathered} H_0:\mu =7 \\ {H}_a:\mu >7 \end{gathered}$$

The z-score is 1.20.

The p-value for the right-tailed test is computed as

$$\begin{gathered} p\left(z>1.20\right)=1-p\left(z⩽1.20\right) \\ =1-0.884 \\ =0.115 \end{gathered}$$

Therefore, the p-value is 0.115.

(b) Calculate

The hypothesis are given by

$$\begin{gathered} H_0:\mu =7 \\ {H}_a:\mu <7 \end{gathered}$$

The z-score is -1.20.

The p-value for the left-tailed test is computed as

$$\begin{gathered} p\left(z<1.20\right)=p\left(z<-1.20\right) \\ =0.884 \end{gathered}$$

Therefore, the p-value is 0.884.

(c) Calculation

The hypothesis are given by

$$\begin{gathered} H_0:\mu =7 \\ {H}_a:\mu \ne 7 \end{gathered}$$

The z-score is 1.20.

Similarly, the p-value for two-tailed tests is computed as

$$\begin{gathered} p value=2\times \varphi \left(-\left|z_{score}\right|\right) \\ =2\times \varphi \left(-\left|1.20\right|\right) \\ =0.230 \end{gathered}$$

Therefore, the p-value is 0.230.