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
Sample Proportion 1Â "\\hat{p_1}=\\dfrac{500}{2000}=0.25"
Favorable Cases 1Â "X_1=200"
Sample Size 1Â "n_1=400"
Sample Proportion 2Â "\\hat{p_2}=\\dfrac{700}{1500}=\\dfrac{7}{15}\\approx0.4667"
Favorable Cases 2Â "X_2=700"
Sample Size 2Â "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)Â 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)}}"
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.
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