Question

question 1 of 10
if a sample proportion is 0.26, which of the following is most likely the range
of possible values that best describes an estimate for the population
parameter?
a. (0.17, 0.39)
b. (0.16, 0.38)
c. (0.15, 0.37)
d. (0.14, 0.36)

Explanation:

Step1: Calculate margin of error for each option

Margin of error \(E\) for a confidence interval around a sample proportion \(\hat{p}\) is calculated as \(E=\hat{p}- \text{lower limit}=\text{upper limit}-\hat{p}\).
For option A: \(E_1 = 0.26 - 0.17=0.09\) and \(0.39 - 0.26 = 0.13\) (not symmetric).
For option B: \(E_2=0.26 - 0.16 = 0.1\) and \(0.38 - 0.26=0.12\) (not symmetric).
For option C: \(E_3=0.26 - 0.15 = 0.11\) and \(0.37 - 0.26 = 0.11\) (symmetric).
For option D: \(E_4=0.26 - 0.14=0.12\) and \(0.36 - 0.26 = 0.1\) (not symmetric).

Step2: Check for symmetric confidence interval

Confidence intervals for proportions are symmetric around the sample proportion \(\hat{p}\) when using the formula \(\hat{p}\pm z\sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}\) (for large - enough \(n\)). We expect the interval to be symmetric around \(\hat{p}=0.26\).

Answer:

C. \((0.15,0.37)\)