91Ó°ÊÓ

Multiple choice: Select the best answer for Exercises 43 to \(46,\) which refer to the following setting. The magazine Sports Illustrated asked a random sample of 750 Division I college athletes, "Do you believe performance- enhancing drugs are a problem in college sports?" Suppose that 30\(\%\) of all Division I athletes think that these drugs are a problem. Let \(\hat{p}\) be the sample proportion who say that these drugs are a problem. The sampling distribution of \(\hat{p}\) is approximately Normal because (a) there are at least 7570 Division I college athletes. (b) \(n p=225\) and \(n(1-p)=525\) (c) a random sample was chosen. (d) a large sample size like \(n=750\) guarantees it. (e) the sampling distribution of \(\hat{\rho}\) always has this shape.

Step by step solution

01

Understand the Question

We need to determine the reason why the sampling distribution of the sample proportion \( \hat{p} \) is approximately normal based on the given options.

02

Recall Conditions for Normality

The sampling distribution of \( \hat{p} \) is approximately normal if certain conditions are met, typically this involves having a large enough sample size to satisfy the conditions \( np \geq 10 \) and \( n(1-p) \geq 10 \).

03

Calculate \( np \) and \( n(1-p) \)

Given \( n = 750 \) and \( p = 0.3 \), calculate \( np \) and \( n(1-p) \): - \( np = 750 \times 0.3 = 225 \)- \( n(1-p) = 750 \times 0.7 = 525 \). Both values are greater than 10.

04

Analyze the Options

Based on our calculations:- (a) at least 7570 athletes is not relevant to the normality.- (b) is true as \( np = 225 \) and \( n(1-p) = 525 \) are both greater than 10.- (c) random sampling ensures unbiasedness, not normality.- (d) a large \( n \) aids in normality but needs \( np \) and \( n(1-p) \).- (e) isn't always true since normal shape depends on \( np \) and \( n(1-p) \).

05

Select the Best Answer

Option (b) is the correct answer as it correctly identifies that both \( np \) and \( n(1-p) \) being greater than 10 ensures the shape is approximately normal.

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Key Concepts

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

Normality Conditions

In statistics, normality conditions are crucial for understanding the behavior of sampling distributions. The sampling distribution of a sample proportion, denoted as \( \hat{p} \), is approximately normal when certain conditions are fulfilled, primarily concerning the sample size.
A key criterion is that both \( np \) and \( n(1-p) \) should be greater than or equal to 10. This ensures that there is a sufficient number of successes and failures in the sample.

• \( np \geq 10 \)
• \( n(1-p) \geq 10 \)

These conditions help in stabilizing the distribution so that it can be approximated by a normal distribution. This approximation is very useful when performing tasks like hypothesis testing or constructing confidence intervals.

Sample Proportion

The sample proportion, denoted as \( \hat{p} \), represents the fraction of items in a sample that possess a certain attribute. In the context of the exercise, it's the proportion of college athletes in the sample who believe performance-enhancing drugs pose a problem.
Calculating \( \hat{p} \) is relatively straightforward. It is simply:\[ \hat{p} = \frac{x}{n} \]Where:

• \( x \) = number of occurrences of the attribute in the sample
• \( n \) = total sample size

Understanding the variability in \( \hat{p} \) is important because the sample proportion may differ from the true population proportion. This is why examining the sampling distribution of \( \hat{p} \) becomes significant, especially under the assumption of normality.
This allows researchers to draw inferences about the population with greater accuracy.

Performance-Enhancing Drugs

Performance-enhancing drugs are substances used by athletes to improve their physical capabilities. The issue of such drugs in sports is quite significant because of ethical, health, and fairness concerns. In the exercise's setting, the magazine seeks to determine how pervasive this problem is in college sports by surveying athletes.
Understanding the perceptions of athletes on this issue can shed light on the severity of the problem, potential deterrents, and the need for policy interventions. This information is essential for:

• Ensuring fair competition
• Protecting athletes' health
• Maintaining the integrity of sports environments

By sampling athletes' views, researchers and policymakers can gauge the extent of awareness and concern regarding performance-enhancing drugs, guiding them to act accordingly.

Random Sampling

Random sampling is a fundamental method in statistics used to ensure that every individual in a population has an equal chance of being selected in a sample. This approach minimizes biases and enhances the representativeness of the sample, making it easier to generalize findings to the larger population.
In the exercise, Sports Illustrated employed random sampling of 750 Division I college athletes to ascertain views on performance-enhancing drugs.
Here are some reasons why random sampling is valuable:

• Increases reliability and validity of results
• Ensures diversity within the sample
• Allows for unbiased estimates of population parameters

By ensuring a random selection, the study achieves a more accurate reflection of the athletes' opinions, mitigating the risks of skewed data due to sampling bias.