Chapter 4, Problem 1E

Suppose that you are testing the following hypotheses where the variance is known:

Find the P-value for the following values of the test statistic.

$\begin{array}{l}{H}_0:\mu =100\\ H_1:\mu \ne 100\end{array}$

(a) Z₀ = 2.75

(b) Z₀ = 1.86

(c) Z₀ = −2.05

(d) Z₀ = −1.86

To determine

Find P-value of the test statistic $Z_0=2.75$.

Answer to Problem 1E

The P-value is 0.0060.

Explanation of Solution

As the alternative hypothesis states that $\mu \ne 100$, the given test is a two-tailed test.

Step-by-step procedure to find P-value using the MINITAB software:

• Choose Graph>Probability Distribution Plot, choose View Probability> OK.
• From the Distribution drop down list box, choose Normal distribution.
• Click the Shaded Area tab.
• Choose the X Value and Both Tails for the region of the curve to be shaded.
• Enter the X Value as 2.75.
• Click OK.
• The output obtained is as follows:

From the output, the P-value at each tail is obtained as 0.002980.

Since the alternative hypothesis (H₁) states that the mean is not equal to 100 (two-tail test), the P-value of the test statistic is calculated by adding the Probability values at both ends of the tail.

The P-value is calculated for two-tailed test is as follows:

$\begin{array}{c}P\text{-Value}=\text{0}\text{.002980}+\text{0}\text{.002980}\\ =0.00596\\ \simeq 0.0060\end{array}$

Thus, the P-value is 0.0060.

To determine

Find P-value of the test statistic $Z_0=1.86$.

Answer to Problem 1E

The P-value is 0.0629.

Explanation of Solution

Step-by-step procedure to find P-value using the MINITAB software:

• Choose Graph>Probability Distribution Plot, choose View Probability> OK.
• From the Distribution drop down list box, choose Normal distribution.
• Click the Shaded Area tab.
• Choose the X Value and Two Tail for the region of the curve to be shaded.
• Enter the X Value as 1.86.
• Click OK.
• The output obtained is as follows:

From the output, the P-value at each tail is obtained as 0.03144.

The P-value is calculated for two-tailed test is as follows:

$\begin{array}{c}P\text{-Value}=0.03144+0.03144\\ =0.06288\\ \simeq 0.0629\end{array}$

Thus, the P-value is 0.0629.

To determine

Find P-value of the test statistic $Z_0=-2.05$.

Answer to Problem 1E

The P-value is 0.0404.

Explanation of Solution

Step-by-step procedure to find P-value using the MINITAB software:

• Choose Graph>Probability Distribution Plot, choose View Probability> OK.
• From the Distribution drop down list box, choose Normal distribution.
• Click the Shaded Area tab.
• Choose the X Value and Two Tail for the region of the curve to be shaded.
• Enter the X Value as -2.05.
• Click OK.
• The output obtained is as follows:

From the output, the P-value at each tail is obtained as 0.02018.

The P-value is calculated for two-tailed test is as follows:

$\begin{array}{c}P\text{-Value}=0.02018+0.02018\\ =0.04036\\ \simeq 0.0404\end{array}$

Thus, the P-value is 0.0404.

To determine

Find P-value of the test statistic $Z_0=-1.86$.

Answer to Problem 1E

The P-value is 0.0629.

Explanation of Solution

Step-by-step procedure to find P-value using the MINITAB software:

• Choose Graph>Probability Distribution Plot, choose View Probability> OK.
• From the Distribution drop down list box, choose Normal distribution.
• Click the Shaded Area tab.
• Choose the X Value and Two Tail for the region of the curve to be shaded.
• Enter the X Value as –1.86.
• Click OK.
• The output obtained is as follows:

From the output, the P-value at each tail is obtained as 0.03144.

The P-value is calculated for two-tailed test is as follows:

$\begin{array}{c}P\text{-Value}=0.03144+0.03144\\ =0.06288\\ \simeq 0.0629\end{array}$

Thus, the P-value is 0.0629.