Overlap of Confidence Intervals In the article “On Judging the...

Question

Overlap of Confidence Intervals In the article “On Judging the Significance of Differences by Examining the Overlap Between Confidence Intervals,” by Schenker and Gentleman (American Statistician, Vol. 55, No. 3), the authors consider sample data in this statement: “Independent simple random samples, each of size 200, have been drawn, and 112 people in the first sample have the attribute, whereas 88 people in the second sample have the attribute.”

c. Use a 0.05 significance level to test the claim that the two population proportions are equal. What do you conclude?

c. Use a 0.05 significance level to test the claim that the two population proportions are equal. What do you conclude?

Accepted Answer

All right, hello, everyone. So this question says, a company wants to compare the proportion of employees who use public transportation to commute in two of its branches. In Branch A, 80 out of 200 employees use public transportation. In Branch B, 110 out of 220 employees use public transportation. At the 0.05 significance level, test whether the proportions are the same in both branches. All right, so first, let's define the information we have. So for branch A, for example, let's say that the sample size or N1 is equal to 200. And X1, which is the number of employees using public transportation, is equal to 80. So for ranch B And 2 is equal to 220. And X2 is equal to 110. So we want to test the null hypothesis. Which in this context would state. That P1 is equal to P2. In other words, that the proportions of employees using public transportation are the same in both branches. Now, the alternative hypothesis on the other hand, would state the opposite, right, that instead these proportions are not equal. So first, let's calculate the sample proportions for each. For branch A, that's Phat one. And that's equal to 80 divided by 200, which gives you 0.4. He had 2 for branch B is 110 divided by 220. Which gives you 0.5. Now, because in this case we're assuming that the null hypothesis is correct, that both proportions are equal, we should use a pooled proportion to combine both samples. So, PHAT. Is equal to The sum of both X values divided by the sum of both sample sizes. So that's The sum of 80 and 110. Divided by the sum of 200. And 220 And so Phat is equal to approximately 0.4524. So now you want to calculate the standard error. And the standard error for the difference in proportions, or SE. Is equal to the square root of Phat. Multiplied by one subtracted by Phat. Multiplied by 1 divided by N1. Added to or excuse me, added to 1 divided by N2. So now, substituting the values obtained earlier, SE is equal to 0.4524. Multiplied by 1 subtracted by 0.4524. And multiplied by 1 divided by 200. Added to 1 divided by 220. So now evaluating this expression gives you approximately 0 point. 0486 And it's with the standard error that we can now calculate the test statistic. So Z Is equal to Phat one. Subtracted by Pha 2 divided by the standard error. So that's equal to 0.4. Subtracted by 0.5. Divided by 0.0486. And this gives you approximately -2.06. So from here, you would find the critical value and make a decision. So, for a two-tailed test at the 0.05 significance level, the critical Z values. Are approximately positive and negative 1.96. Which means that values outside of positive and negative 1.96 are within the rejection region. So notice how in this case, the calculated value of -2.06 is less than -1.96, which means that we can reject the null hypothesis. In contrast, if the test value was between positive and negative 1.96, you would fail to reject. So now this leads us to the final answer, which reads as follows. Reject the null hypothesis. There is sufficient evidence to support the claim that there is a statistically significant difference in proportions. And there you have it. So with that being said, thank you so very much for watching and I hope you found this helpful.

Key Concepts

Confidence Intervals

Hypothesis Testing

Proportions and Sample Size

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