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 95​% confidence interval estimate. The data are shown in the accompanying table. Assuming that the sample data are approximately normal and that the two populations have equal​ variances, develop and interpret the confidence interval estimate. Discuss the result.

Gender_(1=Male_2=Female)	Years_With_the_Club
2	3.5
2	1
1	3
2	2
1	2.5
2	4
2	4.5
1	1
2	2
1	1.5
1	3
2	5.5
1	1.5
1	2.5
1	1
2	3.5
1	3.5
2	1.5
2	1
2	0
1	2.5
1	3.5
2	4.5
1	6
1	2.5
1	5.5
1	2
1	1.5
1	3.5
2	3.5
1	1.5
1	5
2	2
1	5
1	2
2	2
1	2.5
1	0
1	3.5
2	2
2	5.5
1	1
1	2.5
2	1.5
2	1
1	4
2	3
1	1
1	4.5
1	6.5

The 95​% confidence interval for the difference between the two population means for the number of years as members of the fitness club is

nothingless than or equals​(mu 1minusmu 2​)less than or equals

nothing.

​(Round to two decimal places as​ needed.)

What is the interpretation of this​ interval? Select the correct choice below and fill in the answer boxes to complete your choice.

​(Type integers or decimals rounded to two decimal places as needed. Use ascending​ order.)

A.

The interval means that there is​ a(n) ______ probability that the difference between the population​

means, mu 1minusmu 2​, is between ______ and ______ years.

B.

The interval means that the difference between the sample​ means, x overbar 1minusx overbar 2​, will be

Between________ and __________years for ______​% of the samples.

C.

The interval means​ that, with _______​% ​confidence, the difference in mean years that males have been

with the fitness club versus​ females, mu 1minusmu 2​, is between ________ and _______ years.

On the basis of the confidence interval​ produced, can you conclude that a difference exists between the two population means for the number of years as members of the fitness​ club?

A.

Yes​, because the interval contains the value 0.

B.

No​, because the interval contains the value 0.

C.

Yes​, because the interval does not contain the value 0.

D.

No​, because the interval does not contain the value 0.

Answer 1

Part(1).

correct option is C.

The interval means that, with 95% confidence, the difference in mean years that males have been with fitness club versus females, is between-1.1866, 1.5366,

It is given that there are 2 groups one is male and one is female, having diiferent years in club.

And we have to calculate is there any difference between the means(years in club) of these 2 group.

As given in the question that data is normally distributed and assuming there is equal variance.So based on that we can use two sample t-test with equal variance.

Based on the information we can write the null and alternative hypothesis.

    or there is no diiference in the years in club of two groups.

      or There is a difference in years in club between two groups.

From the above table we have:

for males:

   , ,,

for females:

, ,,

the T-value :

where,

: Standard error for differencein means when assuming EQUAL VARIANCE.

Now pluging these values in t- value, we get the t- value:

or

for this test degree of freedom is:

degree of freedom =

df= 30+20-2=48

From t- table now we check the critical value for two tailed at 0.05 significance level.

>

i.e., We cannot reject our null hypothesis, We have to assume that there is no significant difference in the means of two groups, which is that the mean years in club for males and females are approximately equal.

part(2.)

correct option is B. No , because the interval contains the value 0.

The general formula for calculating confidence interval at any confidence interval:

95%CI =

t=2.0

At 95% Confidence interval we have our mean diiference of two group in the range of:

Since the confiidence interval contains the 0. No , because the interval contains the value 0.