Question

Name: Chapter 9: 01. Use the traditional method to test the given hypothesis. Assume that the samples are independent and that they have been randomly selected. In a random sample of 500 people aged 20-24, 22% were smokers. In a random sample of 450 people aged 25-29, 14% were smokers. Test the claim that the proportion of smokers in the two age groups is the same. Use a significance level of 0.01.

Answer 1

Solution :

Null Hypothesis (Claim )

Alternative Hypothesis $H1: P_{1} \neq P_{2}$ (Two tailed test )

p₁ - proportion of smokers in age group 20-24 is = 22% = 0.22

p₂ - proportion of smokers in age group 25 - 29 is = 14% = 0.14

sample sizes are n₁ = 500 and n₂ = 450

Since samples are large, Under H0, the test statistic is

$Z = \frac{p_{1} - p_{2}}{\sqrt{\widehat{P}\widehat{Q}\left ( \frac{1}{n_{1}}+\frac{1}{n_{2}} \right )}}\sim N (0,1)$

where

$\widehat{P} =\frac{n_{1}p_{1}+n_{2}p_{2}}{n_{1}+n_{2}} = \frac{500*0.22+450*0.14}{500+450}= 0.1821$

$\Rightarrow \widehat{Q} =1-\widehat{P} =1 - 0.1821 =0.8179$

0.22 0.14 Z = 3.2046 0.18 0.82 ( 150 500

At 1% level of significance, the critical value of Z is 2.58

Since |Z| = 3.2046 which is greater than 2.58,we reject the null hypothesis and conclude that the proportion of smokers is the two age group is different .