Question

A marketing manager of a Zambian Indigenous Company took a random sample of 2000 customers who bought a newly introduced product and found that 500 customers knew about the product. After a vigorous advertisement, another sample of 1,500 of customers indicated that 700 customers knew about the new product. Test at 5% level of significance if the advertisement increased the number of customers who know about the new product

Answer 1

Sample Proportion 1 $\hat{p_1}=\dfrac{500}{2000}=0.25$

Favorable Cases 1 $X_1=200$

Sample Size 1 $n_1=400$

Sample Proportion 2 $\hat{p_2}=\dfrac{700}{1500}=\dfrac{7}{15}\approx0.4667$

Favorable Cases 2 $X_2=700$

Sample Size 2 $n_2=1500$

The value of the pooled proportion is computed as

$$\bar{p}=\dfrac{X_1+X_2}{n_1+n_2}=\dfrac{500+700}{2000+1500}\approx0.342857$$

Significance Level $\alpha=0.05$

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

$H_0:p_1\geq p_2$

$H_1: p_1<p_2$

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

(2) Rejection Region

Based on the information provided, the significance level is $\alpha=0.05,$ and the critical value for a left-tailed test is $z_c=-1.6449.$

The rejection region for this lrft-tailed test is $R=\{z:z<-1.6449\}.$

The z-statistic is computed as follows:

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$$z=\dfrac{\hat{p}_1-\hat{p}_2}{\sqrt{\bar{p}(1-\bar{p})(1/n_1+1/n_2)}}$$

$$\approx\dfrac{0.25-0.4667}{\sqrt{0.342857(1-0.342857)(1/2000+1/1500)}}$$

$$\approx-13.363891$$

Since it is observed that $z=-13.363891<-1.6449=z_c,$ it is then concluded that the null hypothesis is rejected.

Therefore, there is enough evidence to claim that the population proportion $\hat{p}_1$ is less than $\hat{p}_2,$ at the $\alpha=0.05$ significance level.

Using the P-value approach: The p-value is $p=0,$ and since $p=0<0.05=\alpha,$ it is concluded that the null hypothesis is rejected.

Therefore, there is enough evidence to claim that the population proportion $\hat{p}_1$ is less than $\hat{p}_2,$ at the $\alpha=0.05$ significance level.