In: Statistics and Probability
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.
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. |