Question

In a survey conducted by Yahoo Small Business. 1432 of 1813 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 it is reasonable to regard this sample as representative of adult Americans. Based on these survey data, is it reasonable to conclude that more than three-quarters of adult Americans plan to alter their shopping habits if gas prices remain high?

Short answer

Yes, it's reasonable to conclude that more than three-quarters of adult Americans plan to alter their shopping habits if gas prices remain high.

Step-by-step solution

Identify the Null and Alternative Hypotheses

The Null Hypothesis (\(H_0\)) is that the true proportion (p) of adults planning to alter their habits is equal to 0.75. The Alternative Hypothesis (\(H_a\)) is that true proportion (p) of adults planning to alter their habits is greater than 0.75.

Calculate the sample proportion

Sample proportion (\(\hat{p}\)) is calculated as the number of positive outcomes divided by the sample size. \(\hat{p} = \frac{1432}{1813} = 0.79.\)

Compute the Standard Error

The standard error (SE) is calculated as the square root of the product of the sample proportion, its complement, and the inverse of the sample size. \(SE = \sqrt{ \frac{\hat{p}(1-\hat{p})}{n}} = \sqrt{\frac{0.79(1-0.79)}{1813}} = 0.0098.\)

Calculate the z-score

The z-score is obtained by subtracting the population proportion (p_h0) from the sample proportion (\(\hat{p}\)), and then dividing the result by the standard error (SE). Z = \frac{\hat{p}-p_h0}{SE} = \frac{0.79-0.75}{0.0098} =4.08.

Make a Decision

The critical z-value for a one-tailed test at the 0.05 significance level is 1.645. Because the computed z-value (4.08) is greater than the critical value, there is enough evidence to reject the null hypothesis. Hence, conclude it's reasonable to assert that more than three-quarters of adults plan to change their habits if gas prices remain high.