Question

The article "Men, Women at Odds on Gun Control" (Cedar Rapids Gazette, September 8,1999 ) included the following statement: ' The survey found that 56 percent of American adults favored stricter gun control laws. Sixtysix percent of women favored the tougher laws, compared with 45 percent of men." These figures are based on a large telephone survey conducted by Associated Press Polls. If an adult is selected at random, are the events selected adult is female and selected adult favors stricter gun control independent or dependent events? Explain.

Short answer

Using the calculated probabilities and comparing them, it can be concluded the events 'selected adult is female' and 'selected adult favors stricter gun control' are dependent events, since the probability of both events does not equate to the product of their individual probabilities.

Step-by-step solution

Calculate the Probabilities

First, calculate the overall probability of favoring stricter gun laws, \(P(G)\). According to the survey, 56 percent of adults favor stricter gun control laws, thus \(P(G) = 0.56\). Similarly calculate the probability of the selected adult being female who favors stricter gun control laws, \(P(F \cap G)\). Sixty-six percent of women favor the tougher laws, thus \(P(F \cap G) = 0.66\).

Determine Conditional Probability

Next, determine the conditional probability of an adult favoring stricter gun control laws given they are female, denoted as \(P(G | F)\). In this problem, that equates to the percentage of women who favor stricter gun control laws, which is \(0.66\).

Check for independence

If the events 'selected adult is female' and 'selected adult favors stricter gun control' were independent, the probability of both events occurring would equate to the product of their individual probabilities. In mathematical terms, for events to be independent, \(P(F \cap G) = P(F) \times P(G)\). If this equation does not hold true, then the events are dependent. Therefore, substitute the calculated probabilities to check.

Key concepts

Conditional Probability

Understanding the concept of conditional probability is essential in statistics, as it helps gauge the likelihood of an event occurring given that another event has already taken place. Take our textbook example where the probability of an adult favoring stricter gun control laws given they are female, denoted as \(P(G | F)\), is being calculated. This scenario is a classic case for applying conditional probability.

Conditional probability is calculated using the formula \(P(A | B) = \frac{P(A \cap B)}{P(B)}\), where \(P(A | B)\) is the probability of event A occurring given that event B has occurred, and \(P(A \cap B)\) is the probability of both events A and B occurring. In the context of our survey analysis, knowing that 66% of women support stricter gun laws allows us to understand how gender influences views on this issue.

For individuals new to the concept, practice problems often involve rolling dice, drawing cards, or picking colored balls from a bag under certain conditions. These analogies are helpful in grasping the foundations of conditional probability. Understanding this concept can assist in predicting outcomes and making informed decisions in various real-world scenarios, such as risk assessment and market research.

Dependent Events

Dependent events are pivotal in understanding relationships between different occurrences. In our exercise, investigating whether the selected adult's gender influences their view on gun control leads us to analyze the dependency between two events: being female and favoring stricter gun control laws. If the outcome of one event affects the outcome of another, these events are dependent.

Mathematically, if two events, A and B, are dependent, the probability of both events occurring is not equal to the product of their individual probabilities, which is the rule for independent events: \(P(A \cap B) eq P(A) \times P(B)\). When events are dependent, information about one event alters the likelihood of the other event.

For example, if removing a card from a deck reduces the chances of drawing a second card of the same suit, these events are dependent. In statistical analysis, understanding event dependency is crucial for accurate data interpretation, especially in fields such as healthcare, economics, and social science research.

Statistical Survey Analysis

Statistical survey analysis involves collecting and examining data to uncover trends and insights. In our example, the survey conducted by Associated Press Polls gathered views of American adults on gun control laws. The careful interpretation of such data is fundamental when deducing public opinion on critical issues.

The analysis often starts with determining key metrics, such as percentages of the total population that harbor certain views, followed by an examination of how these views might vary across different demographic groups. These metrics allow for the calculation of both overall probabilities and conditional probabilities, relevant to various subgroups, such as gender.

It's important for survey analysts to consider potential biases in sample selection, survey methodology, and question phrasing, which can significantly affect the results and subsequent conclusions. Students and professionals use statistical surveys to inform policy, business decisions, and even scientific research, making the accuracy and reliability of survey analysis paramount to many fields of study and sectors of industry.