Question

A newspaper reports that the average expenditure on Valentine's Day is $100.89. Do male and female consumers differ in the amounts they spend? The average expenditure in a sample survey of 40 male consumers was $135.67, and the average expenditure in a sample survey of 30 female consumers was $61.64. Based on past surveys, the standard deviation for male consumers is assumed to be $35, and the standard deviation for female consumers is assumed to be $20.

(a)What is the point estimate (in dollars) of the difference between the population mean expenditure for males and the population mean expenditure for females? (Use male − female.)
(b)It 99% confidence, what is the margin of error (in dollars)? (Round your answer to the nearest cent.)
(c)Develop a 99% confidence interval (in dollars) for the difference between the two population means. (Use male − female. Round your answer to the nearest cent.)

Answer 1

The point estimate of the difference between the population mean expenditure for males and females is $74.03. The margin of error is $12.66. The 99% confidence interval for the difference between the two population means is [$61.37, $86.69].

Step-by-step explanation:

(a) The point estimate of the difference between the population mean expenditure for males and females is the difference between the sample means of the two groups. In this case, the point estimate is $135.67 - $61.64 = $74.03.

(b) To calculate the margin of error, we need to find the critical value for a 99% confidence interval. Since the sample sizes are large enough, we can use the Z-distribution. The critical value for a 99% confidence interval is approximately 2.576. The margin of error is then calculated as:

Margin of Error = Critical Value * Standard Error

Standard Error = sqrt[(s1^2/n1) + (s2^2/n2)]

where s1 and s2 are the standard deviations of the male and female populations respectively, and n1 and n2 are the sample sizes of the male and female survey samples.

In this case, the margin of error is approximately 2.576 * sqrt[(35^2/40) + (20^2/30)] = $12.66.

(c) To calculate the confidence interval, we can use the formula:

Confidence Interval = Point Estimate ± Margin of Error

In this case, the confidence interval is $74.03 ± $12.66 = [$61.37, $86.69] (rounded to the nearest cent).