Question

1. alec and ellie are each attempting to estimate the proportion of adults who enjoy reading mystery novels. they each survey a random sample of adults, and, in each sample, the proportion who say they enjoy reading mystery novels is 0.43. alec and ellie proceed to use their sample data to construct a confidence interval. the resulting intervals are shown in the table below. one of these intervals was computed correctly, but the other was not. which interval must be the one that was computed correctly? please explain.

Explanation:

The formula for a confidence interval for a proportion is $\hat{p}\pm z\sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}$, where $\hat{p}$ is the sample proportion. The margin of error $E=z\sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}$ and the interval is symmetric around $\hat{p}$.

For a confidence interval of the form $a$ to $b$, the mid - point of the interval should be $\hat{p}$. The mid - point of an interval $[a,b]$ is calculated as $\frac{a + b}{2}$.

For Alec's interval: $\frac{0.361+0.499}{2}=\frac{0.86}{2}=0.43$.

For Ellie's interval: $\frac{0.342 + 0.478}{2}=\frac{0.82}{2}=0.41
eq0.43$.

Since the sample proportion $\hat{p} = 0.43$ should be the mid - point of the confidence interval (because the confidence interval for a proportion is centered at the sample proportion $\hat{p}$), Alec's interval is computed correctly.

Answer:

Alec's interval (0.361 to 0.499) is computed correctly.