Question

A Gallup survey of 2002 adults found that \(46 \%\) of women and \(37 \%\) of men experience pain daily (San Luis Obispo Tribune, April 6, 2000). Suppose that this information is representative of U.S. adults. If a U.S. adult is selected at random, are the events selected adult is male and selected adult experiences pain daily independent or dependent? Explain.

Short answer

Therefore, the events 'selected adult is male' and 'selected adult experiences pain daily' are not independent but dependent because \(P(Man \cap Pain) \neq P(Man) \times P(Pain)\). Thus, the occurrence of one event does affect the occurrence of the other.

Step-by-step solution

Understand the Data

From the survey, \(46 \%\) of women and \(37 \%\) of men experience pain daily. It is not directly given the proportion of men and women in the population. It's most commonly accepted to assume a \(50-50\) split. As a result, the probability that a randomly selected adult experiences pain daily \(P(Pain)\) would be the average of \(46 \%\) and \(37 \%\), i.e. \(41.5 \%\). Moreover, the probability of selecting a man \(P(Man)\) will be \(50 \%\) from our assumption.

Calculate the Joint Probability

Based on the survey, \(37 \%\) of men experience pain daily. This can be seen as the joint probability of the two events (being male and experiencing pain) and thus, this will be our \(P(Man \cap Pain)\), which is equal to \(37 \% \).

Check for Independence

Two events A and B are independent if \(P(A \cap B) = P(A) \times P(B)\). In our case, we check if \(P(Man \cap Pain)= P(Man) \times P(Pain)\). If this is true, the events are independent. If not, they are dependent. Replacing the values, we need to check if \(37 \% = 50 \% \times 41.5 \%\). After performing the multiplication, we see that \(50 \% \times 41.5 \% = 20.75 \% \) which is not equal to \(37 \% \).

Key concepts

Probability Theory

Probability theory is the branch of mathematics concerned with analyzing random phenomena and modeling uncertainty. It provides a framework for quantifying the likelihood of various outcomes, enabling us to make predictions about events that involve chance.

When a Gallup survey indicates that a certain percentage of a population experiences a specific outcome, such as daily pain, it applies probability theory by presenting the results in terms of percentages—a form of probability. For instance, saying that 46% of women experience pain daily is interpreted as having a probability of 0.46 when a woman is randomly selected from the U.S. adult population.

Understanding such data requires an appreciation for the fundamental rule that the total probability equals one—representing certainty—and the probabilities of all possible outcomes must add up to one. This is instrumental in evaluating survey data which, like the example given, often represents a fraction of a population's experience.

Joint Probability

Joint probability is a measure of two events happening at the same time and is denoted as the probability of the intersection of two events, symbolically represented as \( P(A \cap B) \). It assumes a value between 0 and 1, where 0 indicates that two events never occur together, and 1 implies they always occur simultaneously.

In the survey's context, the joint probability refers to the likelihood of an adult being both male and experiencing pain daily. The calculation of this joint probability is crucial for determining whether the events are independent or dependent. For example, according to the step-by-step solution provided, the joint probability \( P(Man \cap Pain) \) is given as 37%, derived directly from the survey's findings for men.

Survey Data Analysis

Survey data analysis involves processing and interpreting data collected from surveys to draw meaningful conclusions. It can be complex, requiring careful consideration of the survey design, sampling methods, and statistical techniques.

In applying survey data to probability questions, it is essential to ensure the data is representative of the population in question. As suggested in the solution steps provided, sometimes assumptions must be made, such as a 50-50 gender split when specific information is not given. However, it is these very assumptions that can impact the analysis's accuracy, and understanding their effects is crucial in drawing reliable conclusions, especially when inferring the independence or dependence of events in a population.