Question

In 2007, a Gallup poll estimated that 45% of U.S. adults rated their financial situation as “good.” We want to know if the proportion is smaller this year. We gather a random sample of 100 U.S. adults this year and find that 39 rate their financial situation as “good.” Use the output from StatCrunch to complete the following statements about the p-value. Use numbers from the output to fill in the blanks.

The p-value is the probability of observing a sample proportion of (blank) or (Select an answer) in a random sample of size (blank) , when the true population proportion is (blank).

Context [LINK]

Answer 1

Solution:- The p-value is the probability of observing a sample proportion of 0.39 in a random sample of size 100, when the true population proportion is 0.45.

State the hypotheses. The first step is to state the null hypothesis and an alternative hypothesis.

Null hypothesis: P = 0.45
Alternative hypothesis: P < 0.45

Note that these hypotheses constitute a one-tailed test.

Formulate an analysis plan. For this analysis, the significance level is 0.05. The test method, shown in the next section, is a one-sample z-test.

Analyze sample data. Using sample data, we calculate the standard deviation (S.D) and compute the z-score test statistic (z).

S.D = 0.04975
z = (p - P) / S.D

z = - 1.21

where P is the hypothesized value of population proportion in the null hypothesis, p is the sample proportion, and n is the sample size.

Since we have a one-tailed test, the P-value is the probability that the z-score is less than -1.21.

Thus, the P-value = 0.113

Interpret results. Since the P-value (0.113) is greater than the significance level (0.05), we have to accept the null hypothesis.

From the above test we do not have sufficient evidence in the favor of the claim that the proportion is smaller this year.