Question

15.5 percent of a random sample of 1600 under graduates was smokers, whereas 20
percent of a random sample of 900 postgraduates was smokers in a state. Can we
conclude that less number of under graduates are smokers than the postgraduates?​

Answer 1

Yes, we conclude that less number of under graduates are smokers than the postgraduates.

Step-by-step explanation:

We are given that 15.5 percent of a random sample of 1600 under graduates was smokers, whereas 20  percent of a random sample of 900 postgraduates was smokers in a state.

Let $p_1$ = proportion of under graduates who was smokers.

$p_2$ = proportion of post graduates who was smokers.

SO, Null Hypothesis, $H_0$ : $p_1\geq p_2$     {means that more or equal number of under graduates are smokers than the postgraduates}

Alternate Hypothesis, $H_A$ : $p_1< p_2$     {means that less number of under graduates are smokers than the postgraduates}

The test statistics that would be used here Two-sample z-test for proportions;

                           T.S. =  $\frac{(\hat p_1-\hat p_2)-(p_1-p_2)}{\sqrt{\frac{\hat p_1(1-\hat p_1)}{n_1}+\frac{\hat p_2(1-\hat p_2)}{n_2} } }$  ~ N(0,1)

where, $\hat p_1$ = sample proportion of under graduates who was smokers = 15.5%

$\hat p_2$ = sample proportion of post graduates who was smokers = 20%

$n_1$ = sample of under graduates = 1600

$n_2$ = sample of post graduates = 900

So, the test statistics  =  $\frac{(0.155-0.20)-(0)}{\sqrt{\frac{0.155(1-0.155)}{1600}+\frac{0.20(1-0.20)}{900} } }$

                                     =  -2.793

The value of z test statistics is -2.793.

Since, in the question we are not given with the level of significance so we assume it to be 5%. Now, at 5% significance level, the z table gives critical value of -1.645 for left tailed test.

Since, our test statistics is less than the critical value of z as -2.793 < -1.645, so we have sufficient evidence to reject our null hypothesis as it will fall in the rejection region due to which we reject our null hypothesis.

Therefore, we conclude that less number of under graduates are smokers than the postgraduates.