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 = -2.50 is obtained when testing the claim that \(p < 0.75\)

A test statistic value of \(z = - 2.50\) is obtained, and the claim to be tested is \(p < 0.75\).

a.

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

Null Hypothesis: \(p = 0.75\)

Alternative Hypothesis: \(p < 0.75\)

Since there is a lesser than sign in the alternative hypothesis, the test is left-tailed.

b.

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

The z-value is equal to -2.50.

Using the standard normal table, the corresponding left-tailed p-value for z-score equal to -2.50 is equal to:

\(P\left( {z < - 2.50} \right) = 0.0062\)

Thus, the p-value is equal to 0.0062.

To depict the p-value on the standard 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 point “-2.5” on the horizontal axis and then shade the area to the left of the value “-2.5” with blue as shown in the figure.
• Label the shaded region as “p-value = 0.0062”.

The plot below shows the p-value as the shaded area under the standard normal probability graph:

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 0.0062.

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