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Chapter 8: Problem 48

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 gamblingrelated behaviors. \({ }^{17}\) Of the 5594 Division I male athletes in 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 find 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

Expert verified
(a) The confidence level is approximately 99%. (b) Anonymity means the non-respondent bias is unknown, potentially affecting the survey's accuracy.

Step by step solution

01

Calculate the Sample Proportion

First, determine the sample proportion \( \hat{p} \) of male athletes who participated in gambling. The formula for the sample proportion is \( \hat{p} = \frac{x}{n} \), where \( x \) is the number of athletes who gambled, and \( n \) is the total number of male athletes surveyed. Here, \( \hat{p} = \frac{3547}{5594} \approx 0.634 \).
02

Use Margin of Error for Confidence Interval Estimation

The margin of error (ME) is given as 1%. Recall the formula for margin of error in a confidence interval for a proportion: \( ME = z^* \sqrt{\frac{\hat{p}(1-\hat{p})}{n}} \). Here, \( ME = 0.01 \), \( \hat{p} = 0.634 \), and \( n = 5594 \). Solve for \( z^* \).
03

Solve for the Critical Value \( z^* \)

With all the given values, plug them into the margin of error formula to find \( z^* \): \[ 0.01 = z^* \sqrt{\frac{0.634 \times (1 - 0.634)}{5594}} \]Solve this equation to find \( z^* \).
04

Determine the Confidence Level

Once \( z^* \) is calculated, use standard normal distribution tables or software to determine the associated confidence level. Boys solving for approximation, generally morphological enacted % sourced toxicity (by z-source bounced). Summary surrogate is from z-values are 2.576 toward 99 %. Standard applicants, as describes sulfide compound values
05

Interpret Anonymity Impact

Anonymity implies the non-response rate is unknown, potentially skewing results if non-respondents systematically differ from respondents. Interpret results understanding possible changes non-response introduces, for instance weighting discrepancies or demographical profiling errors skew athletes' gambling propensity data.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Confidence Interval
When conducting a survey or study, a confidence interval gives us a range where we expect the true population parameter to lie. This range is constructed around the sample proportion using a confidence level, usually denoted by percentages like 95% or 99%.
A higher confidence level suggests that we are more certain the true parameter lies within our interval, although it typically comes with a wider interval. In the NCAA study, the confidence interval was determined using the margin of error and the sample proportion. This confidence interval tells us that we can be "confident" (at a certain level) that the true proportion of male athletes who gamble falls within this range. The confidence level can be calculated by identifying the critical value from standard normal distribution tables. For instance, a critical value of 2.576 corresponds to a 99% confidence level.
This means we're 99% confident that our calculated interval contains the true proportion of gambling athletes.
Sample Proportion
The sample proportion is a basic statistic used to estimate a population parameter. For this study, the sample proportion, denoted as \( \hat{p} \), is the ratio of the number of male athletes who participated in gambling to the total number surveyed.
The formula is \( \hat{p} = \frac{x}{n} \). Here, \( x = 3547 \) (athletes who gambled) and \( n = 5594 \) (total surveyed), so \( \hat{p} \approx 0.634 \).
This means that approximately 63.4% of the surveyed male athletes reported engaging in gambling.
The sample proportion serves as an estimate of the actual proportion of the entire population (all Division I male athletes), given that the sample is a good representation of the population.
Margin of Error
The margin of error (ME) measures the extent of potential error in our sample estimate when describing the broader population. It represents a range in which the true population parameter is expected to lie, within a given confidence level.
The formula for margin of error in terms of proportion is \( ME = z^* \sqrt{\frac{\hat{p}(1-\hat{p})}{n}} \). In this study, the given margin of error was 1%, or 0.01, meaning our confidence interval was \( \hat{p} \pm 0.01 \).
A small margin of error indicates a more precise estimate of the population parameter. To determine the confidence level, one would typically solve for \( z^* \) in the formula using the provided margin of error. A higher confidence level increases the margin of error, making the interval wider.
Non-response Bias
Non-response bias occurs when individuals who do not respond to a survey differ significantly from those who do, potentially skewing the results.
In the NCAA survey, the anonymity of athletes' responses made it impossible to determine who did not participate, thus the rate of non-respondents is unknown. As a result, any differences in gambling behavior between respondents and non-respondents might lead to bias.
This bias can impact the accuracy of the survey's findings. For instance, if non-responders engage in gambling differently from those who responded, the sample proportion \( \hat{p} \) may not accurately reflect the true population proportion.
  • To mitigate this, researchers can use techniques like follow-up surveys, though anonymity complicates this effort.
  • Understanding and acknowledging the potential impact of non-response bias is crucial for interpreting the results accurately.

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