Question

Please help :(

5. In a simple random sample of 120 women and 150 men from
a population of 150,000, 25% of the women and 37% of the men
are smokers. Suppose you are interested in comparing the
proportion of smokers by sex, do we accept or reject the null
hypothesis of assuming the smoking proportions of women and
men are the same with a .15?

Answer 1

Using the z-distribution, it is found that since the absolute value of the test statistic is greater than the critical value, the null hypothesis is rejected.

At the null hypothesis, we test if the proportions are the same, that is, their subtraction is 0, hence:

$H_0: p_1 - p_2 = 0$

At the alternative hypothesis, it is tested if they are different, that is, their subtraction is not 0, hence:

$H_1: p_1 - p_2 \neq 0$

The proportions and their respective standard errors are given by:

$p_1 = 0.25, s_1 = \sqrt{\frac{0.25(0.75)}{120}} = 0.0395$

$p_2 = 0.37, s_2 = \sqrt{\frac{0.37(0.63)}{150}} = 0.0394$

For the distribution of the difference, the mean and the standard error are given by:

$\overline{p} = p_2 - p_1 = 0.37 - 0.25 = 0.12$

$s = \sqrt{s_1^2 + s_2^2} = \sqrt{0.0395^2 + 0.0394^2} = 0.0558$

The test statistic is given by:

$z = \frac{\overline{p} - p}{s}$

In which $p = 0$ is the value tested at the null hypothesis.

Hence:

$z = \frac{\overline{p} - p}{s}$

$z = \frac{0.12 - 0}{0.0558}$

$z = 2.15$

The critical value for a two-tailed test, as we are testing if two values are different, with a significance level of 0.15 is $|z^{\ast}| = 1.4395$.

Since the absolute value of the test statistic is greater than the critical value, the null hypothesis is rejected.

A similar problem is given at https://brainly.com/question/25676691