Final answer:
To provide a point estimate of the mean annual automobile insurance premium in Michigan, calculate the standard error of the difference between the means using the given data, and compare the z-score to the critical value to determine if the mean cost for auto insurance for teenage boys is greater than that for teenage girls.
Step-by-step explanation:
To provide a point estimate of the mean annual automobile insurance premium in Michigan, we can use the provided data of the random sample. The mean annual cost for 36 teenage boys is $679, while for 23 teenage girls it is $559. To determine whether the mean cost for auto insurance for teenage boys is greater than that for teenage girls, we can compare the two sample means.
First, we can calculate the standard error of the difference between the means using the formula:
SE_diff = sqrt((s1^2 / n1) + (s2^2 / n2))
where s1 and s2 are the standard deviations of the two groups, and n1 and n2 are the sample sizes. In this case, the standard deviation for each group is given as $180. Plugging in the values, we get:
SE_diff = sqrt((180^2 / 36) + (180^2 / 23))
Next, we can calculate the z-score using the formula:
z = (x1 - x2) / SE_diff
where x1 and x2 are the sample means. Plugging in the values, we get:
z = (679 - 559) / SE_diff
Finally, we can compare the z-score to the critical value for a given significance level (e.g., 0.05) to determine whether the mean cost for auto insurance for teenage boys is significantly greater than that for teenage girls.