Answer:
We conclude that the proportion of women who smoke cigarettes is smaller than or equal to the proportion of men at 0.01 significance level.
95% confidence interval for the difference in population proportions of women and men who smoke cigarettes is [0.0062 , 0.1298].
Step-by-step explanation:
We are given that random samples of 125 women and 140 men reveal that 13 women and 5 men smoke cigarettes.
<em>Let </em>
<em> = population proportion of women who smoke cigarettes</em>
<em />
<em> = population proportion of men who smoke cigarettes</em>
So, Null Hypothesis,
:
{means that the proportion of women who smoke cigarettes is smaller than or equal to the proportion of men}
Alternate Hypothesis,
:
{means that the proportion of women who smoke cigarettes is higher than the proportion of men}
The test statistics that will be used here is <u>Two-sample z proportion test</u> <u>statistics</u>;
T.S. =
~ N(0,1)
where,
= sample proportion of women who smoke cigarettes=
=0.104
= sample proportion of men who smoke cigarettes =
= 0.036
= sample of women = 125
= sample of men = 140
So, <u><em>the test statistics</em></u> = ![\frac{(0.104-0.036)-(0)}{\sqrt{\frac{0.104(1-0.104)}{125}+ \frac{0.036(1-0.036)}{140}} }](https://tex.z-dn.net/?f=%5Cfrac%7B%280.104-0.036%29-%280%29%7D%7B%5Csqrt%7B%5Cfrac%7B0.104%281-0.104%29%7D%7B125%7D%2B%20%5Cfrac%7B0.036%281-0.036%29%7D%7B140%7D%7D%20%7D)
= 2.158
Now, at 0.01 significance level, the z table gives critical value of 2.3263 for right tailed test. Since our test statistics is less than the critical value of z as 2.158 < 2.3263, so we have insufficient evidence to reject our null hypothesis due to which <u>we fail to reject our null hypothesis</u>.
Therefore, we conclude that the proportion of women who smoke cigarettes is smaller than or equal to the proportion of men.
<em>Now, coming to 95% confidence interval;</em>
Firstly, the pivotal quantity for 95% confidence interval for the difference between population proportions is given by;
P.Q. =
~ N(0,1)
where,
= sample proportion of women who smoke cigarettes=
=0.104
= sample proportion of men who smoke cigarettes =
= 0.036
= sample of women = 125
= sample of men = 140
<em>Here for constructing 95% confidence interval we have used Two-sample z proportion statistics.</em>
<u>So, 95% confidence interval for the difference between population proportions, </u>
<u> is ;</u>
P(-1.96 < N(0,1) < 1.96) = 0.95 {As the critical value of z at 2.5% level
of significance are -1.96 & 1.96}
P(-1.96 <
< 1.96) = 0.95
P(
<
<
) = 0.95
P(
<
<
) = 0.95
<u>95% confidence interval for</u>
=
[
,
]
= [
,
]
= [0.0062 , 0.1298]
Therefore, 95% confidence interval for the difference in population proportions of women and men who smoke cigarettes is [0.0062 , 0.1298].