Question

A questionnaire collects information from UTS students on gender and whether or not the student smokes. The resultant two-way table is shown below.

Women Men Total

Don’t smoke 153 166 319

Smoke 16 27 43

Total 169 193 362

a) We intend to test whether there is a difference in the proportion of men and women who smoke. Define the parameters, and state the Null and Alternative hypotheses

b) What proportion of men are smokers? What proportion of women are smokers? What is the difference between these proportions?

c) Using information associated with the plots on the next page o Which plot would you use to perform the test in part a)? o Let = 0.05, State your conclusion about whether there is a difference in the proportion of men and women who smoke - with a numerical reference from the appropriate plot

Answer 1

	Women	Men	Total
Do not smoke	153	166	319
Smoke	16	27	43
Total	169	193	362

This is the information that has been provided to us. This is the information collected from UTS students on gender and whether or not the student smokes.

a.

We are required to test if there is a difference in the proportion of men and women who smoke.

Since we have information from a sample, we need to draw inferences about the population from the sample. Thus, we need to if there is a significant difference in the proportion of men and women who smoke. The conclusion cannot be drawn looking at the sample information.

There is variability in the sample which we must account for.

Let us define the parameters:

Let p_m be the proportion of men who smoke; p_m= 27/166= 0.1626506

Let p_w be the proportion of women who smoke; p_w= 16/153= 0.1045752

Let p be the proportion of people who smoke (men and women); p= 43/319= 0.1347962

We will test the hypothesis that the proportions are equal.

The null hypothesis, H₀: p_m p_w

The null hypothesis is that there is no significant difference between the proportion of men and women who smoke.

The alternate hypothesis, H_a: p_mp_w

The alternate hypothesis is that there is in fact a significant difference between the proportion of men and the proportion of women who smoke.

b.

The proportion of men who are smokers (in the sample), p_m= 27/166= 0.1626506

The proportion of women who are smokers (in the sample), p_w= 16/153= 0.1045752

The difference between these proportions, p_m - p_w= 0.1626506- 0.1045752= 0.0580754

c.

The level of significance for the z-test is 0.05 or 5%.

We must first find the z- statistic, and compare this statistic with the z critical value (value from the z-table). This is a two tailed test. Thus,

If the z statistic is greater than or less than the upper limit or the lowere limit (respectively) of the z critical value, then we can reject the null hypothesis, at 5% level of significance. If it is not, we will fail to reject the null hypothesis.

Under the assumption that the null hypothesis is true, the formula for calculating the z- statistic is

Z= (p_m-p_w-0) / sqrt(p(1-p)(1/n_m+ 1/n_w) )

where

p_m= proportion of men who smoke

p_w= proportion of women who smoke

p= proportion of total smokers

n_m= number of men

n_w= number of women.

Thus, Z= (0.0580754-0) / sqrt (0.1347962* (1-0.1347962)(1/169 + 1/193)

= 0.0580754 / sqrt (0.1166262* 0.01109851)

= 1.614217

Thus, the Z critical value is 1.614217

The value from the z-table, for a two tailed test with 5% level of significance is + or - 1.96.

Since our z statistic is less than the z critical value, we fail to reject the null hypothesis, and conclude that there is no significant difference in the proportion of men and the proportion of women who smoke.