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Chapter 4: Problem 19

The National Sporting Goods Association conducted a survey of persons 7 years of age or older about participation in sports activities (Statistical Abstract of the United States: 2002 ). The total population in this age group was reported at 248.5 million, with 120.9 million male and 127.6 million female. The number of participants for the top five sports activities appears here $$\begin{array}{l|cc} & \multicolumn{2}{c} {\text { Participants (millions) }} \\ \text { Activity } & \text { Male } & \text { Female } \\ \hline \text { Bicycle riding } & 22.2 & 21.0 \\ \text { Camping } & 25.6 & 24.3 \\ \text { Exercise walking } & 28.7 & 57.7 \\ \text { Exercising with equipment } & 20.4 & 24.4 \\ \text { Swimming } & 26.4 & 34.4 \end{array}$$ a. For a randomly selected female, estimate the probability of participation in each of the sports activities. b. For a randomly selected male, estimate the probability of participation in each of the sports activities. c. For a randomly selected person, what is the probability the person participates in exercise walking? d. Suppose you just happen to see an exercise walker going by. What is the probability the walker is a woman? What is the probability the walker is a man?

Short Answer

Expert verified
The probability for each activity is calculated by dividing participants by total gender counts. For exercise walking, it's the sum of participants divided by total people. To find gender probabilities, divide gender-specific participants by the total number of exercise walkers.

Step by step solution

01

Understand the Exercise

We need to calculate probabilities for different scenarios based on sports participation data. This involves various probability calculations using gender-specific and total numbers provided in the data table.
02

Calculate Probabilities for Females

To find the probability of participation for females in each activity, divide the number of participating females in each activity by the total number of females, 127.6 million. This will give us:\[P(\text{Bicycle riding | Female}) = \frac{21.0}{127.6},P(\text{Camping | Female}) = \frac{24.3}{127.6},P(\text{Exercise walking | Female}) = \frac{57.7}{127.6},P(\text{Exercising with equipment | Female}) = \frac{24.4}{127.6},P(\text{Swimming | Female}) = \frac{34.4}{127.6}\]
03

Calculate Probabilities for Males

Similarly, to find the probability of participation for males in each activity, divide the number of participating males in each activity by the total number of males, 120.9 million. This will give us:\[P(\text{Bicycle riding | Male}) = \frac{22.2}{120.9},P(\text{Camping | Male}) = \frac{25.6}{120.9},P(\text{Exercise walking | Male}) = \frac{28.7}{120.9},P(\text{Exercising with equipment | Male}) = \frac{20.4}{120.9},P(\text{Swimming | Male}) = \frac{26.4}{120.9}\]
04

Calculate Probability for Exercise Walking

The probability that a randomly selected person participates in exercise walking is calculated by dividing the total number of exercise walking participants (both male and female) by the total population of 248.5 million:\[P(\text{Exercise walking}) = \frac{28.7 + 57.7}{248.5}\]
05

Calculate Conditional Probabilities for Exercise Walker's Gender

a) The probability that an exercise walker is a woman is given by the ratio of female exercise walkers to total exercise walkers:\[P(\text{Woman | Exercise walking}) = \frac{57.7}{28.7 + 57.7}\]b) The probability that an exercise walker is a man is given by the ratio of male exercise walkers to total exercise walkers:\[P(\text{Man | Exercise walking}) = \frac{28.7}{28.7 + 57.7}\]
06

Verification and Interpretation

Verify calculations for accuracy and ensure each probability is between 0 and 1. Interpret the results to understand which groups are more active in each sport.

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Key Concepts

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

Probability
Probability is a fundamental concept in statistics that measures the likelihood of an event occurring. In the context of sports participation, probability can help us understand how likely it is for individuals of a certain gender to participate in specific sports activities. Probability values, also known as probabilities, range from 0 to 1, where 0 indicates impossibility and 1 indicates certainty. Calculating probabilities involves dividing the number of successful outcomes by the total number of possible outcomes.

In this exercise, we use probability to estimate the likelihood of people participating in various sports activities based on provided data. For example, to find the probability of a randomly selected female bicycling, we divide the number of female bicyclists by the total female population. This simple formula can be applied to each category, helping us derive insights about sports preferences among genders.
Sports Participation
Sports participation refers to the involvement of individuals in various sports and physical activities. Understanding participation trends can offer insights into public health, fitness levels, and potential areas for improvement in sports programs. Sports such as bicycle riding, camping, exercise walking, exercising with equipment, and swimming are listed with participation figures for both genders. This information helps identify which sports are more popular among different demographics.

Factors influencing sports participation may include cultural preferences, availability of facilities, and promotion of sporting events. By examining participation rates, we can assess the popularity of different activities and target efforts to increase involvement in less popular sports. This can lead to enhanced health benefits for a broader segment of the population.
Gender Analysis
Gender analysis in statistical studies examines differences in participation or outcomes between males and females. Through gender analysis, we gain insights into how gender influences sports preferences and participation rates. In the survey, data shows distinct patterns such as more females participating in exercise walking, while more males engage in camping.

Conducting gender analysis helps in the formulation of effective strategies to balance participation across genders. For instance, understanding which activities appeal more to certain genders can aid in designing programs that encourage balanced gender participation. This is crucial for achieving equality in sports, ensuring all individuals have access to the physical and social benefits of sports activities.
Statistical Analysis
Statistical analysis is the process of collecting and interpreting data to discover underlying patterns and trends. It involves various methods of calculating figures like averages, probabilities, and distributions. In this exercise, statistical analysis is used to determine the probabilities of participation across different sports and genders. This requires calculating ratios, which involve dividing the number of participants in a specific group (e.g., females in swimming) by the total number of individuals in that group (e.g., total females).

Through statistical analysis, we can also calculate conditional probabilities, which provide insights into the likelihood of another event occurring given a known condition, such as determining the probability of an exercise walker being male or female. Proper execution of statistical analysis ensures accuracy and comprehensiveness in data interpretation, allowing us to derive actionable insights from raw numbers.

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Most popular questions from this chapter

The U.S. Bureau of Labor Statistics collected data on the occupations of workers 25 to 64 years old. The following table shows the number of male and female workers (in millions) in each occupation category (Statistical Abstract of the United States: 2002 ). $$\begin{array}{lrr} \text { Occupation } & \text { Male } & \text { Female } \\ \text { Managerial/Professional } & 19079 & 19021 \\ \text { Tech./Sales/Administrative } & 11079 & 19315 \\ \text { Service } & 4977 & 7947 \\ \text { Precision Production } & 11682 & 1138 \\ \text { Operators/Fabricators/Labor } & 10576 & 3482 \\ \text { Farming/Forestry/Fishing } & 1838 & 514 \end{array}$$ a. Develop a joint probability table. b. What is the probability of a female worker being a manager or professional? c. What is the probability of a male worker being in precision production? d. Is occupation independent of gender? Justify your answer with a probability calculation.

Assume that we have two events, \(A\) and \(B\), that are mutually exclusive. Assume further that we know \(P(A)=.30\) and \(P(B)=.40\) a. What is \(P(A \cap B) ?\) b. What is \(P(A | B) ?\) c. A student in statistics argues that the concepts of mutually exclusive events and independent events are really the same, and that if events are mutually exclusive they must be independent. Do you agree with this statement? Use the probability information in this problem to justify your answer. d. What general conclusion would you make about mutually exclusive and independent events given the results of this problem?

Small cars get better gas mileage, but they are not as safe as bigger cars. Small cars accounted for \(18 \%\) of the vehicles on the road, but accidents involving small cars led to 11,898 fatalities during a recent year (Reader's Digest, May 2000 ). Assume the probability a small car is involved in an accident is .18. The probability of an accident involving a small car leading to a fatality is .128 and the probability of an accident not involving a small car leading to a fatality is .05. Suppose you learn of an accident involving a fatality. What is the probability a small car was involved? Assume that the likelihood of getting into an accident is independent of car size.

The prior probabilities for events \(A_{1}, A_{2},\) and \(A_{3}\) are \(P\left(A_{1}\right)=.20, P\left(A_{2}\right)=.50,\) and \(P\left(A_{3}\right)=\) \(.30 .\) The conditional probabilities of event \(B\) given \(A_{1}, A_{2},\) and \(A_{3}\) are \(P\left(B | A_{1}\right)=.50\) \(P\left(B | A_{2}\right)=.40,\) and \(P\left(B | A_{3}\right)=.30\) a. Compute \(P\left(B \cap A_{1}\right), P\left(B \cap A_{2}\right),\) and \(P\left(B \cap A_{3}\right)\) b. Apply Bayes' theorem, equation (4.19), to compute the posterior probability \(P\left(A_{2} | B\right)\). c. Use the tabular approach to applying Bayes' theorem to compute \(P\left(A_{1} | B\right), P\left(A_{2} | B\right)\) and \(P\left(A_{3} | B\right)\)

The prior probabilities for events \(A_{1}\) and \(A_{2}\) are \(P\left(A_{1}\right)=.40\) and \(P\left(A_{2}\right)=.60 .\) It is also known that \(P\left(A_{1} \cap A_{2}\right)=0 .\) Suppose \(P\left(B \text { ? } A_{1}\right)=.20\) and \(P\left(B | A_{2}\right)=.05\) a. Are \(A_{1}\) and \(A_{2}\) mutually exclusive? Explain. b. Compute \(P\left(A_{1} \cap B\right)\) and \(P\left(A_{2} \cap B\right)\) c. Compute \(P(B)\) d. Apply Bayes' theorem to compute \(P\left(A_{1} | B\right)\) and \(P\left(A_{2} | B\right)\).
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