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.