Question

1. a random sample of 250 sports fans are surveyed, and 62% of these surveyed fans say that football is their favorite sport. if the goal is to use this sample data to construct a confidence interval, what will the margin of error be?

a. 0.031
b. 0.060
c. 0.063
d. 0.050
e. its impossible to answer this question.

Explanation:

Step1: Identify the formula for margin of error

The formula for margin of error \(E\) for a proportion is \(E = z\sqrt{\frac{p(1 - p)}{n}}\). When the confidence level is not given, we assume a \(95\%\) confidence level. For a \(95\%\) confidence level, \(z=1.96\). Here, \(p = 0.62\) (since \(62\%=0.62\)) and \(n = 250\).

Step2: Calculate \(1-p\)

\(1 - p=1 - 0.62 = 0.38\)

Step3: Calculate \(\frac{p(1 - p)}{n}\)

\(\frac{p(1 - p)}{n}=\frac{0.62\times0.38}{250}=\frac{0.2356}{250}=0.0009424\)

Step4: Calculate \(\sqrt{\frac{p(1 - p)}{n}}\)

\(\sqrt{0.0009424}\approx0.0307\)

Step5: Calculate the margin of error \(E\)

\(E=1.96\times0.0307\approx0.060\)

Answer:

B. \(0.060\)