Answer to Question #233189 in Statistics and Probability for Sos

Question

6 After a recent A I DS awareness campaign, the department of National Health commissioned a market research company to conduct a survey on its effectiveness. Their brief was to establish whether the recall rate of teenagers differed from that of young adults (20 30 years of age). The market research company interviewed a random sample of 640 teenagers and 420 young adults. It was found that 362 teenagers and 260 young adults were able to recall the A I DS awareness slogan of “A I DS: don’t let it happen”. The management wants to test, at the 5% level of significance, the null hypothesis H0 : There is an equal recall rate between teenagers and young adults. The test statistic is

Answer 1

For sample of teenagers , we have that the sample proportion is

$$\hat{p}_1=\dfrac{X_1}{N_1}=\dfrac{362}{640}=0.5656.$$

For sample of young adults, we have that the sample proportion is

$$\hat{p}_2=\dfrac{X_2}{N_2}=\dfrac{260}{420}=0.6190.$$

The value of the pooled proportion is computed as 

$$\dfrac{X_1+X_2}{N_1+N_2}=\dfrac{362+260}{640+420}=0.5868$$

The following null and alternative hypotheses for the population proportion needs to be tested:

$H_0:p_1=p_2$

$H_1:p_1\not=p_2$

This corresponds to a two-tailed test, and a z-test for two population proportions will be used.

Based on the information provided, the significance level is $\alpha=0.05,$ and the critical value for a two-tailed test is $z_c=1.96.$

The rejection region for this two-tailed test is $R=\{z:|z|>1.96\}$

The z-statistic is computed as follows:

$$z=\dfrac{\hat{p}_1-\hat{p}_2}{\sqrt{\bar{p}(1-\bar{p})(\dfrac{1}{N_1}+\dfrac{1}{N_2})}}$$

$$=\dfrac{0.5656-0.6190}{\sqrt{0.5868(1-0.5868)(\dfrac{1}{640}+\dfrac{1}{420})}}\approx-1.7269$$

Since it is observed that $|z|=1.7269<1.96=z_c,$ it is then concluded that the null hypothesis is not rejected.

Using the P-value approach: The p-value is $p=2P(Z<-1.7269)=0.084186,$ and since $p=0.084186>0.05=\alpha,$ it is concluded that the null hypothesis is not rejected.

Therefore, there is not enough evidence to claim that the population proportion $p_1$ is different than $p_2,$ at the $\alpha=0.05$ significance level.