Question

Researchers examined all sports-related concussions reported to an emergency room for children ages 5 to 18 in the United States over the course of one year. \({ }^{11}\) Table 2.7 displays the number of concussions in each of the major activity categories. (a) Are these results from a population or a sample? (b) What proportion of concussions came from playing football? (c) What proportion of concussions came from riding bicycles? (d) Can we conclude that, at least in terms of concussions, riding bicycles is more dangerous to children in the US than playing football? Why or why not? $$\begin{array}{l|r}\hline \text { Activity } & \text { Frequency } \\ \hline \text { Bicycles } & 23,405 \\ \text { Football } & 20,293 \\\\\text { Basketball } & 11,507 \\ \text { Playground } & 10,414 \\\\\text { Soccer } & 7,667 \\\\\text { Baseball } & 7,433 \\\\\text { All-Terrain Vehicle } & 5,220 \\\\\text { Hockey } & 4,111 \\\\\text { Skateboarding } & 4,408 \\\\\text { Swimming } & 3,846 \\\\\text { Horseback Riding } & 2,648 \\ \hline \text { Total } & 100,952 \\\\\hline\end{array}$$

Short answer

(a) Population. (b) 0.201. (c) 0.232 . (d) No, because the proportion of concussions alone does not consider other important factors.

Step-by-step solution

Identify Population or Sample

The results contain sports-related concussions from throughout the United States for children ages 5 to 18 over the course of an entire year. Thus, these results represent a population.

Compute The Proportion of Concussions from Football

The proportion of concussions from football can be found by dividing the frequency of football concussions, 20293, by the total number of concussions, 100952. Therefore, the proportion is \(\frac{20293}{100952} = 0.201.\)

Compute The Proportion of Concussions from Bicycles

The proportion of concussions from riding bicycles is computed by dividing the frequency of bicycle concussions by the total number of concussions. Therefore, the proportion is \(\frac{23405}{100952} = 0.232.\)

Compare Risks from Football and Bicycles

Although bicycle-related concussions account for a higher proportion of total concussions (0.232) than football-related concussions (0.201), we cannot conclude that riding bicycles is more dangerous than playing football. This is because the proportion alone does not take into account other variables such as the number of children participating in each activity, duration of their participation, and the nature of each activity.

Key concepts

Population vs Sample

When conducting a study in statistics, it's crucial to understand the difference between a 'population' and a 'sample'.

A population comprises all members of a specified group, while a sample is a subset of the population that is used to represent the entire group. In the context of our exercise, the researchers are looking at the entirety of sports-related concussions reported to an emergency room for a specific age group in the United States, over the course of one year. Since the data encompasses all events meeting the criteria nationwide for that period, it's representative of a population rather than a sample.

When a population is too large to measure every single member, researchers take a sample. The importance of this distinction lies in how the results are generalized. Results from a population can tell us about the entire group with certainty, while results from a sample can only provide estimates with a certain level of confidence.

Concussion Proportion Calculation

To understand the impact of different activities on sports-related concussions, we must calculate the proportion of concussions resulting from each activity. The proportion is a type of ratio that represents a part of a whole.

The formula to calculate the proportion of concussions from a specific activity is \[\begin{equation}Proportion = \frac{Number\ of\ Concussions \ from\ Activity}{Total\ Number\ of\ Concussions}\end{equation}\]

In the exercise, we're given the numbers for both football and bicycles. To get the proportion of football-related concussions, divide the number of football concussions (20,293) by the total number of concussions (100,952), which gives us a proportion of approximately 0.201. Similarly, for bicycle-related concussions, divide 23,405 by 100,952 to get a proportion of about 0.232. These calculations help us understand the relative frequency of concussions from these activities within the context of the total data set.

Conducting Statistical Comparisons

Interpreting statistical data properly is a vital skill, especially when attempting to compare risks between activities as in our exercise.

To determine if one activity is more dangerous than another, merely comparing the proportions of concussions isn't enough. This is because proportions don't account for variables such as exposure time, the number of participants, and the intensity of the activity.

For a more accurate comparison, researchers need to conduct a statistical test that incorporates these factors. For instance, injury rates might be standardized per a certain number of hours of activity or participant. If the data shows that per 1,000 hours of participation, a higher rate of concussions is reported for bicycling than football, we might then consider bicycling to be more dangerous with respect to concussions.

Additionally, statistical comparisons must consider the possibility of confounding variables and biases. Researchers often use control groups and multiple variables to ensure that the conclusions drawn are as accurate as possible.