The average expenditure on Valentines Day was expected to be

The average expenditure on Valentine\'s Day was expected to be $100.89 (USA Today, February 13, 2006). Do male and female consumers differ in the amounts they spend? The average expenditure in a sample survey of 50 male consumers was $135.67, and the average expenditure in a sample survey of 30 female consumers was $68.64. Based on past surveys, the standard deviation for male consumers is assumed to be $39, and the standard deviation for female consumers is assumed to be $19.

What is the point estimate of the difference between the population mean expenditure for males and the population mean expenditure for females (to 2 decimals)?

At 99% confidence, what is the margin of error (to 2 decimals)?

Develop a 99% confidence interval for the difference between the two population means (to 2 decimals).
( , )

Solution

Given X₁ =135.67      n1 = 50

The null hypothesis is

Against the alternative hypothesis

The test statistic t is given by

Z = (X₁- X₂)/((s₁)²/n1)+((s₂)²/n2) N_(0,1)

Z = (135.67 -687.64)/ ((39)²/50)+((19)²/30) N_(0,1)

Z = 10.2876

The table value for is given by Z_tab=2.58

Therefore Z_cal>Z_tab i.e., we reject the null hypothesis

Hence the population mean expenditure for males is not equal to the population mean expenditure for females

standard Error (X₁- X₂) = ((s₁)²/n1)+((s₂)²/n2)

standard Error (X₁- X₂) = ((39)²/50)+((19)²/30) = 6.5156

99% confident limit for µ₁ = µ₂ i.e., the difference between the population mean for males and population mean for females are

X₁- X₂ ± 2.58 S.E(X₁- X₂)

67.03±2.58(6.6156)

[50.2198 , 83.8403 ]