Question

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: $$ \begin{array}{l|cc} \hline & \text { Male } & \text { Female } \\ \hline \text { Uses Seat Belts Regularly } & .10 & .175 \\ \begin{array}{l} \text { Does Not Use Seat Belts } \\ \text { Regularly } \end{array} & .40 & .325 \\ \hline \end{array} $$ Assume that these proportions are representative of adults in the United States and that a U.S. adult 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.

Short answer

a. \(0.275\) b. \(0.10\) c. \(0.325\) d. \(0.448\) e. No, they are not equal, given that the proportions of male and female who do not use seat belts regularly are different.

Step-by-step solution

Identify the total proportions

First, observe that the total proportion of adults is 1. The proportions of different groups (seat belts users and non-users) among males and females don’t add up to 1, hence, it implies that the remaining proportions make up for the other gender.

Calculate the probability for part a

The probability that the selected adult regularly uses a seat belt is given by the sum of the proportions of males and females who regularly use a seat belt. This can be calculated as \(0.10 + 0.175 = 0.275\).

Calculate the probability for part b

The probability that the selected adult regularly uses a seat belt given that he is male is directly given in the table. It is \(0.10\).

Calculate the probability for part c

The probability that the selected adult does not use a seat belt regularly given that she is female is given directly in the table. It amounts to \(0.325\).

Calculate the probability for part d

Here, we want to find the probability that the individual is female given that she does not use a seat belt regularly. To calculate this, we find the proportion of females who do not use a seat belt regularly and divide it by the total proportion of adults who do not use seat belts regularly. This can be calculated as \(\frac{0.325}{0.325 + 0.40} = 0.448\).

Evaluate if the probabilities from parts c and d are equal

Comparing the probabilities calculated in steps 4 and 5, it can be seen that they are not equal. This is because seat belt usage is not equally distributed among genders. The proportions of male and female who do not use seat belts differ, thus leading to different probabilities.

Key concepts

Seat Belt Usage Survey

When discussing the topic of seat belt usage, it's vital to consider how such data is collected and interpreted. A seat belt usage survey, like the one mentioned in the USA Today article from June 6, 2000, involves collecting data from a sample of individuals and using it to draw conclusions about the entire population's behavior. For the survey to provide valuable insights, the sample must be representative, meaning it should reflect the diverse characteristics of the entire population.

For instance, if the survey concludes that 10% of adult men and 17.5% of adult women regularly use seat belts, these figures give us an estimation of seat belt usage within these respective groups across the entire nation. However, the reliability of these numbers hinges on the survey's ability to capture a wide range of individuals from different demographics—age, income, location—which ensures that the findings are not biased toward a particular group's behavior.

Conditional Probability

Conditional probability is a fundamental concept in statistics that measures the probability of an event occurring given that another event has already occurred. It's like asking what the chances are of finding a red apple in a basket knowing that you're only looking at the apples and not the oranges.

In the seat belt usage context, when you want to know the likelihood of a randomly selected male adult using a seat belt regularly, gender becomes a condition of the probability. Mathematically, this is represented as P(Uses Seat Belt | Male), and from our survey data, it was given as 10%. This differs from the overall probability of using a seat belt because it only considers one part of the population—men. Understanding conditional probability is crucial for making precise conclusions about specific segments of a population.

Representative Proportions

Representative proportions are percentages that accurately reflect the characteristics of a larger group. When conducting surveys or studies, it is essential that the proportions found in the collected data are indicative of the broader population to ensure valid results. If the proportions are skewed, any inference drawn from the data might lead to incorrect conclusions or interpretations.

In our seat belt survey, the proportions of adults who regularly use seat belts are assumed to be representative of the entire U.S. adult population. This assumption is necessary for part a of the exercise where we calculate the probability of a randomly selected adult being a regular seat belt user by summing the percentages of both genders, resulting in a combined usage rate of 27.5%. Without representative proportions, our understanding of seat belt usage among all adults in the U.S. could be inaccurate.