Question

In: Statistics and Probability

Given in the table are the BMI statistics for random samples of men and women. Assume...

Given in the table are the BMI statistics for random samples of men and women. Assume that the two samples are independent simple random samples selected from normally distributed​ populations, and do not assume that the population standard deviations are equal. Complete parts​ (a) and​ (b) below. Use a 0.01 significance level for both parts.

Male BMI Female BMI
m m1 m2
n 47 47
x 27.0582 26.2309
s 7.742635 4.732567

a. Test the claim that males and females have the same mean body mass index​ (BMI).

What are the null and alternative​ hypotheses?

b. The test​ statistic =

c. The​ P-value =

d. Construct a confidence interval suitable for testing the claim that males and females have the same mean BMI.

e. Does the confidence interval support the conclusion of the​ test?

f. State the conclusion for the test.

Solutions

Expert Solution

Male BMI Female BMI
n 47 47
Sample mean 27.0582 26.2309
Sample SD 7.74264 4.732567

We are testing whether the population means are equal or not. For this we have been given the data follows normal but the population SD are unknown. So we can use the t-dist for unequal population variances.

Where v1 = n1 - 1 and v2 = n2 -1

p-value = 2P( > Test Stat)

Where ' P( > Test Stat)' is found using t-dsit tables with above formula for df.

Test

Since we are testing the claim whether they are equal or not it is two sided.

Null: The mean BMI are equal for males and females.

Alternative: The mean BMI are not equal for males and females.

Denominator
2 Test Stat 0.62502
3 df 76.162
p-value  (P(t(76) > 0.63)= 0.2669 0.53383
Assuming = 0.05
Since p-value > 0.05
Decision We do not reject the null hypothesis
Conclusion

There is insufficient evidence to conclude that the means are not equal.

For the interval we have the following

Where the alpha = 0.05

C.V. =

We find this using t-dist tables.

4 Confidence interval
t-critical 1.99167
Lower L -1.80897
Upper L. 3.46357

e. Does the confidence interval support the conclusion of the​ test?

Since the interval includes '0', that is the null difference, we do not reject the null hypothesis at 5%. This supports the same conclusion as the test.

f. State the conclusion for the test.

Since p-value > 0.05
Decision We do not reject the null hypothesis
Conclusion

There is insufficient evidence to conclude that the means are not equal.


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