Question

In order to estimate the difference between the yearly incomes of marketing managers in the East and West of the United States, the following information was gathered.

East West
n₁ = 40 n₂ = 45
= 72 (in $1,000) = 78 (in $1,000)
s₁ = 6 (in $1,000) s₂ = 8 (in $1,000)

a. Develop an interval estimate for the difference between the average yearly incomes of the marketing managers in the East and West. Use a = 0.05.

Answer 1

The interval estimate for the difference between the average yearly incomes of marketing managers in the East and West is calculated using a two-sample t-interval formula, incorporating the sample means, standard deviations, sizes, and the t-value from the t-distribution for the given confidence level.

Step-by-step explanation:

To estimate the difference between the average yearly incomes of marketing managers in the East and West of the United States, we can use a two-sample t-interval. To calculate this interval, we need the sample means, sample standard deviations, and sample sizes for each region. The formula for the confidence interval is given by:

(Øx₁ - Øx₂) ± t* √((s₁²/n₁) + (s₂²/n₂))

Where Øx₁ and Øx₂ are the sample means, s₁ and s₂ are the sample standard deviations for the East and West respectively, n₁ and n₂ are the sample sizes, and t* is the t-value from the t-distribution for the desired confidence level and degrees of freedom calculated using the Welch-Satterthwaite equation.

Using the data provided (East: n₁ = 40, Øx₁ = 72,000, s₁ = 6,000; West: n₂ = 45, Øx₂ = 78,000, s₂ = 8,000), and a confidence level of 95% (a = 0.05), we first calculate the degrees of freedom and the t-value (not provided, usually found in t-distribution tables or using statistical software), then plug these values into the formula to obtain the interval estimate for the difference in average incomes.