Question

Gambling and the NCAA Gambling is an issue of great concern to those involved in college athletics. Because of this concern, the National Collegiate Athletic Association (NCAA) surveyed randomly selected student athletes concerning their gambling-related behaviors. Of the $5594$Division $I$male athletes who responded to the survey, $3547$reported participation in some gambling behavior. This includes playing cards, betting on games of skill, buying lottery tickets, betting on sports, and similar activities. A report of this study cited a $1\%$margin of error.

a. The confidence level was not stated in the report. Use what you have learned to

estimate the confidence level, assuming that the NCAA took an SRS.

b. The study was designed to protect the anonymity of the student athletes who responded. As a result, it was not possible to calculate the number of students who were asked to respond but did not. How does this fact affect the way that you interpret the results?

Short answer

a. Confidence level is $87.88\%$

b. Non Response Bias.

Step-by-step solution

Given Information

It is given that $n=5594$

$x=3547$

$E=1\%=0.01$

Calculation of Confidence Level

Sample Proportion is calculated as:

$\hat{p}=\frac{x}{n}$

$\hat{p}=\frac{3547}{5594}=0.6341$

Margin of error is calculated as:

$E=Z_{\alpha /2}\times \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$

$0.01=Z_{\alpha /2}\times \sqrt{\frac{0.6341(1-0.6341)}{5594}}$

$\Rightarrow Z_{a/2}=\frac{0.01}{0.00644}=1.55$

Hence, Probability:

$P(-1.55<Z<1.55)=P(Z<1.55)-P(Z<-1.55)$

$P(-1.55<Z<1.55)=0.9394-0.0606$

$P(-1.55<Z<1.55)=0.8788=87.88\%$

Hence, confidence level is$87.88\%$

Bias in Study

Non Sampling Bias are:

Selection Bias, Response Bias, Non Response Bias.

If we don't have data for everybody in sample, it is result of non response bias.

So, we cannot generalize the result as we don't have sufficient data.