Q1-17BSC - Elementary Statistics

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 \(p > 0.3\)

Step-by-Step Solution

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 1: Given Information

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

Step 2: 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.

Step 3: 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:

 

Step 4: 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.