Question

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 = 1.00 is obtained when testing the claim that \(p > 0.3\)

Short answer

a. The test is right-tailed.

b. The p-value is equal to 0.1587.

c. The decision of the statistical test is to fail to reject \({H_0}\).

Step-by-step solution

Given Information

A test statistic value of \(z = 1.00\) is obtained, and the claim to be tested is \(p > 0.3\).

Identify the hypotheses and tail of the test

a.

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

Null Hypothesis: \(p = 0.3\)

Alternative Hypothesis: \(p > 0.3\)

Where p is the population proportion.

Since there is greater than sign in the alternative hypothesis, the test is right-tailed.

P-value

b.

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

The z-value is equal to 1.00.

Using the standard normal table, the corresponding right-tailed p-value for z-score equal to 1.00 is equal to:

\(\begin{array}{c}P\left( {z > 1.00} \right) = 1 - P\left( {z < 1.00} \right)\\ = 1 - 0.8413\\ = 0.1587\end{array}\)

Thus, the p-value is equal to 0.1587.

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

• Plot a horizontal axis representing the z-score. Alos 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 “1” on the horizontal axis and then shade the area to the right of the value “1” with blue as shown in the figure.
• Label the shaded region as “p-value = 0.1587”.

The following plot shows the probability value (p-value) as the shaded area under the normal probability graph:

Decision about the test

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 0.1587.

Since the p-value is greater than 0.05, so the decision is to fail to reject the null hypothesis.