Question

1. blane and steph are each attempting to estimate the proportion of adults who make online purchases at least once per month. they each survey a different random sample of adults. in each sample, \\( \hat { p } = 0.682 \\). blane and steph each use their sample data to construct a confidence interval. blanes interval is from 0.623 to 0.739 and stephs interval is from 0.627 to 0.737. one of these intervals was calculated correctly, but the other was not. which interval must be the one that was calculated incorrectly?

a. stephs interval must have been calculated incorrectly.
b. its impossible to answer this question without knowing the sample sizes.
c. its impossible to answer this question without knowing the level of confidence used for each interval.
d. its impossible to answer this question without knowing the sample sizes or the level of confidence used for each interval.
e. blanes interval must have been calculated incorrectly.

Explanation:

Step1: Calculate the mid - point of Blane's interval

The mid - point of an interval \((a,b)\) is \(\frac{a + b}{2}\). For Blane's interval \((0.623,0.739)\), the mid - point is \(\frac{0.623+0.739}{2}=\frac{1.362}{2}=0.681\).

Step2: Calculate the mid - point of Steph's interval

For Steph's interval \((0.627,0.737)\), the mid - point is \(\frac{0.627 + 0.737}{2}=\frac{1.364}{2}=0.682\).

Since \(\hat{p}=0.682\) should be the mid - point of the confidence interval (because the formula for a confidence interval for a proportion is \(\hat{p}\pm z\sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}\)), Blane's interval has a mid - point that is not equal to \(\hat{p}\) while Steph's interval has a mid - point equal to \(\hat{p}\).

Answer:

E. Blane’s interval must have been calculated incorrectly.