Question

In a survey conducted by Yahoo Small Business, 1,432 of 1,813 adults surveyed said that they would alter their shopping habits if gas prices remain high (Associated Press, November 30,2005\() .\) The article did not say how the sample was selected, but for purposes of this exercise, assume that the sample is representative of adult Americans. Based on the survey data, is it reasonable to conclude that more than threequarters of adult Americans would alter their shopping habits if gas prices remain high?

Short answer

The explanation of the outcome completely depends on the Z score calculated in step 3. If the calculated Z score is greater than 1.645 (Z > 1.645), then that will be an evidence to reject the null hypothesis. In this case, it would be reasonable to conclude that more than 75% of American adults would alter their shopping habits if gas prices remain high. However, if the calculated Z score is less than or equal to 1.645 (Z ≤ 1.645), then we would not reject the null hypothesis and the conclusion would be that there is not enough evidence to support the claim.

Step-by-step solution

Statement of Hypotheses

Set up the null hypothesis (\(H_0\)) and the alternative hypothesis (\(H_1\)). Let p represent the population proportion of American adults who would change their shopping habits if gas prices stay high. Here, \(H_0: p \leq 0.75\) (less than or equal to threequarters) and \(H_1: p > 0.75\) (more than threequarters)

Calculation of Sample Proportion

Calculate the sample proportion (\(\hat{p}\)). The sample proportion is calculated as the number of successes divided by the sample size. Here, successes refers to adults who said they would change their shopping habits, which is 1,432 of a sample size of 1,813. Therefore, \(\hat{p} = \frac{1432}{1813} = 0.790\)

Test Statistic Calculation

Calculate the test statistic which follows a normal distribution, since we assume the sample is representative of adult Americans. This is done using the formula for the Z score in hypothesis testing for proportions: \( Z = \frac{(\hat{p}-p_0)}{\sqrt{(p_0(1-p_0))/n}} \). Here \(p_0\) is the assumed population proportion under the null hypothesis, \(\hat{p}\) is the sample proportion and n is the sample size. Plugging the numbers, we get \( Z = \frac{(0.790-0.75)}{\sqrt{(0.75)(0.25)/1813}} \)

Rejection Region and Conclusion

The test is one-tail and at a 0.05 significance level. So, we reject the null hypothesis if Z > 1.645. Using a Z-table or calculator, we calculate Z and compare it to 1.645. We reject the null hypothesis if Z > 1.645, meaning that we will conclude it is reasonable to believe more than three-quarters of adults would alter their shopping habits if calculated Z is bigger than 1.645.