91Ó°ÊÓ

A random sample of 185 college soccer players who had suffered injuries that resulted in loss of playing time was made with the results shown in the table. Injuries are classified according to severity of the injury and the condition under which it was sustained. $$ \begin{array}{|c|c|c|c|} \hline & \text { Minor } & \text { Moderate } & \text { Serious } \\ \hline \text { Practice } & 48 & 20 & 6 \\ \hline \text { Game } & 62 & 32 & 17 \\ \hline \end{array} $$ a. Give a point estimate for the proportion \(p\) of all injuries to college soccer players that are sustained in practice. b. Construct a \(95 \%\) confidence interval for the proportion \(p\) of all injuries to college soccer players that are sustained in practice. c. Give a point estimate for the proportion \(p\) of all injuries to college soccer players that are either moderate or serious.

Step by step solution

01

Calculate Total Injuries

First, we'll calculate the total number of injuries by adding all injuries from both practice and game situations across all severity levels.\[\text{Total Injuries} = 48 + 20 + 6 + 62 + 32 + 17 = 185 \]

02

Point Estimate for Practice Injuries

The point estimate for the proportion of injuries sustained during practice is calculated by dividing the total number of practice injuries by the total number of injuries.\[\text{Proportion of Practice Injuries} = \frac{48 + 20 + 6}{185} = \frac{74}{185} \approx 0.40 \]

03

Construct 95% Confidence Interval

To construct a 95% confidence interval for the proportion of practice injuries, we use the formula for the confidence interval of a proportion: \[p \pm z \times \sqrt{\frac{p(1-p)}{n}} \]Where \(p\) is the sample proportion of practice injuries, \(z\) is the z-score corresponding to a 95% confidence level (approximated to 1.96), and \(n\) is the total number of injuries.\[p = \frac{74}{185} \approx 0.40 \ CI = 0.40 \pm 1.96 \times \sqrt{\frac{0.40 \times (1-0.40)}{185}} \ CI = 0.40 \pm 1.96 \times 0.0362 \ CI = 0.40 \pm 0.0709 \ \text{CI} = [0.3291, 0.4709] \]

04

Point Estimate for Moderate or Serious Injuries

The point estimate for the proportion of injuries that are either moderate or serious is calculated by dividing the total number of moderate and serious injuries by the total number of injuries.\[\text{Proportion of Moderate or Serious Injuries} = \frac{20 + 6 + 32 + 17}{185} = \frac{75}{185} \approx 0.405 \]

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Key Concepts

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

Proportion Estimation

Proportion estimation is a fundamental concept in statistics used to determine the fraction of items in a population that satisfy a particular condition.
For instance, if we want to know what proportion of all injuries sustained by soccer players occur in practice, proportion estimation becomes our tool.

This involves calculating the ratio of injuries under consideration to the total number of recorded injuries.

• In our exercise, we're focusing on practice injuries. Thus, the proportion of practice injuries is calculated as the sum of all practice injuries divided by the total injuries.
• Proportion estimation is crucial in understanding how different categories (like practice versus game) compare within a dataset.

The broader application extends beyond sports or injuries, allowing statisticians and researchers to grasp various phenomena across endless subjects.

Random Sample

A random sample is a subset of individuals chosen from a larger population, ensuring that every member of the population has an equal chance of being selected.
This is critical in statistics as it helps in obtaining unbiased and representative data.

In the context of our exercise:

• We have a random sample of 185 college soccer players who suffered injuries, ensuring that the collected data reflects the diversity in injuries among all such players.
• Random sampling minimizes biases that could potentially skew the research results.

By employing random samples, researchers can make more reliable predictions and generalizations about the whole population based on their analysis of the sample.

Point Estimate

A point estimate is a single-valued approximation for a population parameter that is unknown but is being estimated.
It is derived directly from the sample data and provides a specific number which is then used for further statistical analysis.

In the exercise at hand, we calculate:

• The proportion of injuries occurring in practice sessions as a point estimate: dividing the number of practice injuries by the total injuries results in a figure of approximately 0.40.
• The proportion of moderate or serious injuries, yielding a point estimate of about 0.405.

Point estimates are straightforward, but they do not express the uncertainty inherent in sampling. Therefore, they are usually accompanied by intervals (like confidence intervals) to give a range for the actual parameter value.

95% Confidence Interval

A 95% confidence interval gives us a range in which we can say with 95% certainty that the true population parameter lies within.
The process involves calculating a margin of error around the point estimate, which provides insights into the accuracy of our estimate.

In calculating a 95% confidence interval:

• We use the formula for a confidence interval of a proportion: \[ p \pm z \times \sqrt{\frac{p(1-p)}{n}} \]
• In our exercise, this gives a confidence interval of approximately [0.3291, 0.4709] for the practice injuries, meaning the true proportion of practice injuries lies within this range with 95% confidence.

This confidence level is commonly chosen in statistics, balancing the trade-off between certainty and precision.

Injury Classification

Injury classification categorizes injuries by severity, helping to analyze and understand the nature of incidents athletes face.
This is critical because different types of injuries (e.g., minor versus serious) require varying responses and prevention strategies.

For instance, within the exercise:

• Injuries are divided into minor, moderate, and serious, providing a clear understanding of the severity distribution among injured players.
• This classification aids in the identification of patterns or trends, like whether more severe injuries occur during games rather than practice.

Injury classification is essential for tailoring medical responses and developing preventive measures within sports contexts and beyond.