Question

A study has been conducted on the rate of depression and
their relation to demographic features such as age, race, gender,
etc. The survey was administered to 155 patients and it was found
women are more likely to be depressed compared to men (Data
extracted from: Gottlieb SS, Khatta M, Friedmann E, et al. The influence
of age, gender, and race on the prevalence of depression in
heart failure patients. J Am Coll Cardiol. 2004; 43(9):1542-1549.
doi:10.1016/j.jacc.2003.10.064.). The following data related to the
depression and the gender has been imported from the study:

Depressed Not Depressed Total
Men 54 68 122
Women 21   12 33
Total 75   80 155

a. Set up the null and alternative hypotheses to determine whether
there is a difference in the depression levels of men and women.
b. At 0.05 significance level, compute x2 STAT. Is there any evidence
of a significant difference between the proportion of men
and women and their depression levels?
c. Determine the p-value in (a) and interpret its meaning.

Answer 1

	Depression
Gender	yes	no	Total
Male	54	68	122
Female	21	12	33
Total	75	80	155

This is test for difference of proportion. where

x: no. of depressed men and women

We have '1' for men and '2' for women

Binomial diff

= 54/ 75 = 0.72      = 68 / 80= 0.85

= 0.7871

Test

(a) : Proportion of depression levels is same in men and women.

: Proportion of depression levels is different in men and women.

b) Test stat : ............Where the null difference = 0

After sub the values we get

  test stat = -1.9758

c) p-value = 2 P(Z >|Test Stat|)

= 2 P( Z > 1.98)

=2 * (1 - P(Z < 1.98))

= 2 *( 1 - 0.9759)

  p-value = 0.048

Since p-value < 0.05 (level of significance)

We reject the null hypothesis and conclude that significantly the depression levels is different for men and women.

p-value is the probability of null hypothesis being true. Therefore the probability that the proportion is same for levels of men and women is 0.048. (which is lower than acceptance level of 0.05)