Final answer:
To determine if the mean price at a Kansas City dealership is different from the national average, a two-tailed hypothesis test using a t-test is conducted with null hypothesis μ = $10,192 and alternative hypothesis μ ≠ $10,192. Type I error occurs if a false positive is concluded, while Type II error occurs if a true difference is not detected.
Step-by-step explanation:
The hypothesis test to determine if the mean price of used cars from a Kansas City dealership is statistically different than the national average involves setting up a two-tailed test with the following null (H₀) and alternative (H₁) hypotheses:
- H₀: μ = $10,192 (The population mean price at the dealership is equal to the national mean)
- H₁: μ ≠ $10,192 (The population mean price at the dealership is not equal to the national mean)
A two-tailed test is appropriate because the manager wants to know if the prices are either higher or lower than the national mean. To conduct this test, one would typically use a t-test for the sample mean because we are dealing with a sample of less than 30 sales and assuming we do not know the population standard deviation.
The test would involve calculating the t-statistic using the sample mean, sample standard deviation, and sample size to determine if the observed difference is statistically significant. Type I and Type II errors would be defined as follows:
- Type I error: Concluding that the dealership's mean price is different from the national mean when it is not (false positive).
- Type II error: Failing to detect a difference when in fact the dealership's mean price is different from the national mean (false negative).
If the p-value obtained from the t-test is less than the significance level (α), usually set at 0.05, the null hypothesis is rejected, suggesting a statistically significant difference between the dealership's mean price and the national mean price.