Final answer:
A hypothesis test reveals that given a sample of 30 fatal accidents where 14 are due to driver error, there is not sufficient evidence to reject the null hypothesis at the α = 0.05 level. Therefore, the claim by the American Automobile Association that driver error accounts for 54 percent of fatal auto accidents cannot be statistically disproven based on this sample.
Step-by-step explanation:
To verify if the American Automobile Association's (AAA) claim that driver error accounts for approximately 54 percent of all fatal auto accidents is accurate, a hypothesis test is conducted.
Given that out of 30 randomly selected fatal accidents, 14 were found to be caused by driver error, we want to determine if this sample proportion significantly differs from the AAA's stated proportion of 0.54. Using the significance level (α) of 0.05, we can perform a two-proportion z-test.
The hypothesis for our test are:
Null Hypothesis: p = 0.54 (The proportion is equal to the AAA's proportion)
Alternative Hypothesis: p ≠ 0.54 (The proportion is not equal to the AAA's proportion)
To conduct the z-test, first calculate the sample proportion (p-hat), standard error (SE), and z-score:
p-hat = 14/30 = 0.4667
SE = sqrt((p*(1-p))/n) = sqrt((0.54*(1-0.54))/30) = 0.0912
z-score = (p-hat - p)/SE = (0.4667 - 0.54)/0.0912 = -0.8035
Next, we need to check this z-score against the z-table to find the p-value. A z-score of about -0.80 gives us a p-value greater than 0.05, which means we fail to reject the null hypothesis. There is not enough evidence to conclude that the proportion of fatal accidents caused by driver error is significantly different from the AAA proportion.