The average expenditure on Valentine\'s Day was expected to be $100.89 (USA Toda

Question

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 40 male consumers was $135.67, and the average expenditure in a sample survey of 32 female consumers was $68.64. 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 $16.

A) 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)?

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

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

Explanation / Answer

a)

Calculating the means of each group,              
              
X1 =    135.67          
X2 =    68.64          

hence,

X1 - X2 = 67.03 [ANSWER]

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b)
              
Calculating the standard deviations of each group,              
              
s1 =    35          
s2 =    16          
              
Thus, the standard error of their difference is, by using sD = sqrt(s1^2/n1 + s2^2/n2):              
              
n1 = sample size of group 1 =    40          
n2 = sample size of group 2 =    32          
Thus, df = n1 + n2 - 2 =    70          
Also, sD =    6.214901447          
              
For the   0.99   confidence level, then      
              
alpha/2 = (1 - confidence level)/2 =    0.005          
t(alpha/2) =    2.647904624          

Hence,

Margin of error = t(alpha/2) * sD = 16.45646628 [ANSWER]

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c)

Thus,

lower bound = [X1 - X2] - t(alpha/2) * sD =    50.57353372          

upper bound = [X1 - X2] + t(alpha/2) * sD =    83.48646628          
              
Thus, the confidence interval is                              
(   50.57353372   ,   83.48646628   ) [ANSWER]

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Hi! If you use another method/formula in calculating the degrees of freedom in this t-test, please resubmit this question together with the formula/method you use in determining the degrees of freedom. That way we can continue helping you! Thanks!

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