91Ó°ÊÓ

USA Today (June 6,2000 ) gave information on seat belt usage by gender. The proportions in the following table are based on a survey of a large number of adult men and women in the United States. $$ \begin{array}{ccc} & \text { Male } & \text { Female } \\ \hline \text { Uses Seat Belts Regularly } & .10 & .175 \\ \text { Does Not Use Seat Belts } & .40 & .325 \\ \text { Regularly } & & \\ \hline \end{array} $$ Assume that these proportions are representative of adult Americans and that an adult American is selected at random. a. What is the probability that the selected adult regularly uses a seat belt? b. What is the probability that the selected adult regularly uses a seat belt given that the individual selected is male? c. What is the probability that the selected adult does not use a seat belt regularly given that the selected individual is female? d. What is the probability that the selected individual is female given that the selected individual does not use a seat belt regularly? e. Are the probabilities from Parts (c) and (d) equal? Write a couple of sentences explaining why this is so.

Step by step solution

01

Overall Seat Belt Usage

Adding the proportions of the male and female users who regularly wear seat belts will give us the total probability of an individual using the seat belt regularly. P(Use Seat Belts Regularly) = P(Male usage) + P(Female usage) = 0.10 + 0.175 =0.275.

02

Seat Belt Usage by Males

To find the probability that the selected individual regularly uses a seat belt given that individual is male, P(Seat belt usage | Male), we simply look at the proportion given in the table for the male category: 0.10.

03

Non-usage of Seat Belt by Females

To find the probability that a selected individual does not use a seat belt given the individual is female P(Non-usage | Female), we refer to the proportion in the table for the female category who do not use seat belts regularly: 0.325.

04

Female non-usage given No Seat Belt Usage

To find the probability that a selected individual is female given that they do not use a seat belt regularly P(Female | Non-usage), we have to use conditional probability. This will be equal to the proportion of women that do not use a seat belt divided by the total population that do not use a seat belt = 0.325/(0.40 + 0.325) = 0.325/0.725 ≈ 0.4482758620

05

Comparing Probabilities in Part c and Part d

Comparing probabilities obtained in part c (P(Non-usage | Female) = 0.325) and part d (P(Female | Non-usage) ≈ 0.4482758620), it's clear they aren't equal. The former is the probability of a female not using a seatbelt, while the latter is the probability that a randomly chosen person who doesn't use a seatbelt is female. These two conditions don't reflect the same event and consequently, the probabilities are also different. We need to be clear about the total population considered while calculating probabilities. Respectively they are 'all females' and 'all people who don't use seat belts' and this is why the probabilities differ.

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Key Concepts

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

Seat Belt Usage

Seat belt usage is a key topic when discussing road safety and statistical probability. Understanding how different populations use seat belts can highlight significant safety trends, which is crucial for developing awareness and policy. In our example, the focus is on the seat belt habits of adult men and women in the U.S. This kind of data helps us understand broader patterns in a population.

Here, the data tells us that 10% of surveyed men consistently use seat belts, while the figure is 17.5% for women. These statistics were obtained from a survey and are assumed to represent the general habits of adult Americans. Surveys like this provide insights into behaviors that may not be uniformly distributed across gender lines. This could shape the implementation of targeted seat belt campaigns better suited to each gender's characteristics.

For students studying statistics, seat belt usage data exemplifies how real-world behaviors can be interpreted using mathematical tools, offering a practical application of probability theories.

Conditional Probability

Conditional probability forms the backbone of many statistical analyses. It deals with finding the probability of an event occurring, given another event has already occurred. In simpler terms, it answers questions like, "What is the probability of event A, given that event B is known to have occurred?"

In our exercise, we calculated the conditional probability that a selected adult uses a seat belt given they are male. Here, this was simply the proportion from our data: 0.10, or 10%. Conditional uses of information like this are powerful because they allow us to isolate variables and understand specific interactions between them.

Two concepts often confused are conditional probability and joint probability — the latter being the chance of both events happening together. Clarifying this key distinction can reduce errors in interpretation. Understanding and mastering conditional probabilities allow better predictions and finer data analysis.

Gender Statistics

Gender statistics involve analyzing data separately for men and women to uncover patterns and inequalities. These analyses help in understanding the different experiences and behaviors of different gender groups.

The exercise presented gives us a simple but meaningful example. It shows us how a specific behavior, such as seat belt usage, varies between genders. This statistic reveals that a significantly higher percentage of women regularly use seat belts compared to men (17.5% vs. 10%).

These insights are critical for creating gender-sensitive policies. For instance, if a public health initiative aims to increase seat belt usage, such gender-based statistics guide where efforts might be more needed. Moreover, gender statistics also highlight necessary discussions on why these differences exist, opening the floor for cultural, social, or psychological explorations. Students learning about statistics will come to value these distinctions for both theoretical study and practical analysis.