Question

At a baseball game, $42$of $65$randomly selected people own an iPod. At a rock concert occurring at the same time across town, $34$of $52$randomly selected people own an iPod. A researcher wants to test the claim that the proportion of iPod owners at the two venues is different. A $90\%$confidence interval for the difference (Game − Concert) in population proportions is $(-0.154,0.138)$. Which of the following gives the correct outcome of the researcher’s test of the claim?

a. Because the confidence interval includes $0$, the researcher can conclude that the proportion of iPod owners at the two venues is the same.

b. Because the center of the interval is $-0.008$, the researcher can conclude that a higher proportion of people at the rock concert own iPods than at the baseball game.

c. Because the confidence interval includes $0$, the researcher cannot conclude that the proportion of iPod owners at the two venues is different.

d. Because the confidence interval includes more negative than positive values, the researcher can conclude that a higher proportion of people at the rock concert own iPods than at the baseball game.

e. The researcher cannot draw a conclusion about a claim without performing a significance test.

Short answer

Option (c) Because the confidence interval includes $0$, the researcher cannot conclude that the proportion of iPod owners at the two venues is different. is the correct outcome of the researcher’s test of the claim.

Step-by-step solution

Given information

We need to find correct outcome of the researcher’s test of the claim.

Simplify

The confidence interval is as follows:

$(-0.154,0.138)$

Now, we see that the confidence interval contains zero, which suggests that the population proportions are very likely to be equal, and thus we cannot determine that they are different.
As a result, option (c) is the proper choice; the researcher cannot conclude that the fraction of iPod owners at the two venues is different because the confidence interval comprises zero.