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The hypothesis are given as follows:
Null hypothesis: [tex]H_0: \pi_1 = \pi_2[/tex]Alternative hypothesis: [tex]H_1: \pi_1 \neq \pi_2[/tex]The test statistic is given as follows:
z = 38.92.
The critical value is given as follows:
z = 1.96.
The conclusion is given as follows:
Since the test statistic is greater than the critical value, reject the null hypothesis. There is sufficient evidence to conclude that there is a significant difference between the proportion of Superbanked and the Unbanked with respect to using credit cards.
The p-value is of: 0.
The p-value is the probability that in a random sample of 1,000 Superbanked consumers and 1,000 Unbanked customers, the difference between the proportions of Superbanked and Unbanked customers is at least 0.65.
What are the hypothesis tested?At the null hypothesis, it is tested if the hypothesis are equal, that is:
[tex]H_0: \pi_1 = \pi_2[/tex]
At the alternative hypothesis, it is tested if they are different, that is:
[tex]H_1: \pi_1 \neq \pi_2[/tex]
We are testing if the two proportions are different, hence we have a two-tailed test, using the z-distribution, with a significance level of 0.05, and the critical value is given as follows:
z = 1.96.
Then the decision rule is given as follows:
|z| < 1.96: do not reject the null hypothesis.|z| > 1.96: reject the null hypothesis.How to obtain the test statistic and p-value?To obtain the test statistic, we need to obtain the proportion and the standard error for each sample, as follows:
[tex]\pi_1 = 0.89, s_1 = \sqrt{\frac{0.89(0.11)}{1000}} = 0.0099[/tex][tex]\pi_2 = 0.24, s_1 = \sqrt{\frac{0.24(0.76)}{1000}} = 0.0135[/tex]Hence, for the distribution of differences, the mean and the standard error are given as follows:
p = 0.89 - 0.24 = 0.65.s = square root (0.0099² + 0.0135²) = 0.0167.The test statistic is given by the division of the mean by the standard error, hence:
z = p/s = 0.65/0.0167 = 38.92.
Using a z-distribution calculator, with a two-tailed test and z = 38.92, the p-value is of 0.
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