Question

Dr. Fictitious Jones conducts a study to test whether losing weight reduces cholesterol levels. He tests 15 people in January. Six months later, after they have all participated in a weight loss program and lost at least 3% of their starting body weight, he measures them again.

a. What is your null hypothesis?
b. What is your research hypothesis?
c. What test would you run to test your hypothesis?
d. Run the test at the 5% level of significance.
e. State whether you accept or reject the null hypothesis.
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Answer 1

a.
Let $\mu_b$ and $\mu_a$ be the mean cholesterol level before and after the weight loss.
Null hypothesis H0: Mean cholesterol is same before and after losing the weight. That is, $\mu_a$ = $\mu_b$ or, $\mu_b$ - $\mu_a$ = 0

b. Alternative hypothesis H0: Mean cholesterol is lower after weight loss as compared with before weight loss. That is, $\mu_a$ < $\mu_b$ or $\mu_b$ - $\mu_a$​​​​​​​ > 0

c.
Since, the data is paired data where both measurements (before and after weight loss) was conducted on the same person, we will use matched-pairs t-test.

d.
The difference in cholesterol level before and after is,

-5 15 0 10 5 -15 15 -10 0 40 -15 5 40 -5 -5

Mean difference in cholesterol level before and after, m = 5

Standard deviation of cholesterol level before and after, s = 17.0084

Standard error of mean difference = 17.0084 / $\sqrt{15}$ = 4.39155

Test statistic, t = Mean Difference / Standard error = 5 / 4.39155 = 1.14

Degree of freedom = n-1 = 15 - 1 = 14

P-value = P(t > 1.14) = 0.1367

e.

Since, p-value is greater than the significance level of 0.05, we fail to reject (accept) null hypothesis H0, and conclude that there is no significant evidence that mean cholesterol is lower after weight loss as compared with before weight loss.