P-Values. In Exercises 17–20, do the following:

a. Identify the hypothesis test as being two-tailed, left-tailed, or right-tailed.

b. Find the P-value. (See Figure 8-3 on page 364.)

c. Using a significance level of \(\alpha \) = 0.05, should we reject \({H_0}\)or should we fail to reject \({H_0}\)?

The test statistic of z = 2.01 is obtained when testing the claim that \(p \ne 0.345\).

A test statistic value of \(z = 2.01\) is obtained, and the claim to be tested is \(p \ne 0.345\).

a.

In correspondence with the given claim, the following hypotheses are set up:

Null Hypothesis: \(p = 0.345\)

Alternative Hypothesis: \(p \ne 0.345\)

Since there is a not equal sign in the alternative hypothesis, the test is two-tailed.

b.

The test statistic to test the given claim is the z-value.

The z-value is equal to 2.01.

Using the standard normal table, the corresponding two-tailed p-value for z-score equal to 2.01 is equal to:

\(\begin{aligned}{c}2 \times P\left( {z < 2.01} \right) = 2 \times \left( {0.0222} \right)\\ = 0.0444\end{aligned}\)

Thus, the p-value is equal to 0.0444.

To depict the p-value on the normal probability graph, follow the given steps:

• Plot a horizontal axis representing the z-score. Also, label it as “z-score”.
• Sketch a bell-shaped curve and draw a vertical dotted line corresponding to the value “0” on the horizontal axis
• Mark the points “-2.01” and “2.01” on the horizontal axis and then shade the area to the left of the value “-2.01” and to the right of the value “2.01” with blue as shown in the figure.
• Label the two shaded regions as “p-value = 0.0222”.

The following plot shows the probability value (p-value) as the shaded area under the normal probability graph. Here, the sum of the two p-values corresponds to the requited two-tailed p-value (0.0444).

c.

If the p-value is less than the level of significance, the null hypothesis is rejected; otherwise, not.

Here, the level of significance is equal to 0.05, and the p-value is equal to 0.0444.

Since the p-value is less than 0.05, the null hypothesis is rejected.