Question

the owner of a fitness center is interested in estimating the difference in mean years that female members have been with the club compared with male members. he wishes to develop a 99% confidence interval estimate. The data are showen in the accompanying table. Assuming that the same data are approximately normal and that the two populations have equal variances, develop and interpret the confidence interval estimate. dicuss the results

gender
1=male 2=female
1
2
2
2
1
2
2
2
1
2
1
2
1
2
2
1
2
1
2
2
1
1
2
2
2 2 2 2 1 2 2 2 2 2 2 1 2 1 2 2 2 2 2 1 1 2 1 2 1 2 years with the club 3.5 1 2.5 2 3.5 4 5.5 1 2 1.5 3 6.5 1.5 3 1 3.5 4.5 1.5 1 0 3 3.5 4.5 6 3.5 5.5 2 1.5 3 3.5 1.5 5.5 2 3 2 2 2.5 0 2.5 2 5 1 2.5 1.5 1 4.5 2.5 1 4.5 7 show formulas in excel

Answer 1

For finding out confidence interval first we have to calculate mean and standard deviation of years male and female are with gym. We have used excel to find out mean and standard deviation.

Given x??, s?, n?, x??, s?, n?, and Level of Confidence:

x?? (sample mean 1) = 2.4688 x?? (sample mean 2) = 3

s? (sample standard deviation 1) = 1.1757 s? (sample standard deviation 2) = 1.8586

n? (sample size 1) = 16 n? (sample size 2) = 34

Degrees of Freedom = 16+34 - 2 = 48

Confidence Interval = 99%, So ? = 0.01

?/2 = 0.01/2 = 0.005

t_?/2 = 2.6822 (Obtained using excel formula =TINV(0.01,48))

Standard Error = ?[(n?-1)(s?)² + (n?-1)(s?)²]/[n? + n? - 2] • ?1/n? + 1/n?
= ?2.806855375625 • ?0.09191176470588236
= 0.5079

Lower Bound = (x?? - x??) - t_?/2•(?[(n?-1)(s?)² + (n?-1)(s?)²]/[n? + n? - 2]•?1/n? + 1/n?)
= (2.4688 - 3) - (2.6822)(0.5079)
= -1.8935

Upper Bound = (x?? - x??) + t_?/2•(?[(n?-1)(s?)² + (n?-1)(s?)²]/[n? + n? - 2]•?1/n? + 1/n?)
= (2.4688 - 3) + (2.6822)(0.5079)
= 0.8311

Confidence Interval of difference between male and female mean years with gym = (-1.8935, 0.8311)

Interpretation:

?? = Population mean for males

?? = Population mean for females

Since we do not know if the confidence interval (-1.8935, 0.8311) contains (?? - ??) or not, we are only 99% confident that (-1.8935, 0.8311) contains (?? - ??).