A random sample of 20 customers who drive luxury cars showed that their average distance driven between oil changes was 3187 miles with a sample standard deviation of 42.50 miles. Another sample of 10 customers who drive compact lower-price cars resulted in an average distance of 3214 miles with a sample standard deviation of 50.70 miles. Suppose that the standard deviations for the two populations are not equal. Using the 5% significance level, can you conclude that the mean distance between oil changes is lower for all luxury cars than for all compact lower-price cars?

Question

A random sample of 20 customers who drive luxury cars showed that their average distance driven between oil changes was 3187 miles with a sample standard deviation of 42.50 miles. Another sample of 10 customers who drive compact lower-price cars resulted in an average distance of 3214 miles with a sample standard deviation of 50.70 miles. Suppose that the standard deviations for the two populations are not equal. Using the 5% significance level, can you conclude that the mean distance between oil changes is lower for all luxury cars than for all compact lower-price cars?

Answer 1

Use same process shown in previous post.

Answer 2

To answer this question, we can conduct a two-sample t-test to compare the means of the two samples. The assumption is that the populations from which the samples are drawn are normally distributed.

The null hypothesis (H0) states that there is no difference in the mean distance between oil changes for luxury cars and compact lower-price cars. The alternative hypothesis (Ha) states that the mean distance between oil changes for luxury cars is lower than that for compact lower-price cars.

We can set the significance level at 0.05 or 5%.

To perform the t-test, we need the following information:

Sample 1:
Sample mean (x̄1) = 3187 miles
Sample standard deviation (s1) = 42.50 miles
Sample size (n1) = 20

Sample 2:
Sample mean (x̄2) = 3214 miles
Sample standard deviation (s2) = 50.70 miles
Sample size (n2) = 10

Now, we can calculate the test statistic (t-value) using the formula:

t = (x̄1 - x̄2) / √[ (s1^2 / n1) + (s2^2 / n2) ]

Plugging in the values:

t = (3187 - 3214) / √[ (42.50^2 / 20) + (50.70^2 / 10) ]

Calculating this expression gives us the t-value.

Once we have the t-value, we can compare it to the critical value from the t-distribution tables with degrees of freedom (df) equal to (n1 + n2 - 2).

If the calculated t-value is less than the critical value, we can reject the null hypothesis (H0) and conclude that there is evidence that the mean distance between oil changes is lower for all luxury cars than for all compact lower-price cars.

On the other hand, if the calculated t-value is greater than the critical value, we fail to reject the null hypothesis and conclude that there is not enough evidence to suggest that the mean distance between oil changes is lower for luxury cars.

Performing these calculations will provide us with the final answer to the question.