91Ó°ÊÓ

In a recent Harris Poll, a random sample of adult Americans (18 years and older) was asked, "Given a choice of the following, which one would you most want to be?" Results of the survey, by gender, are given in the contingency table. $$\begin{array}{lcccccc} & \text { Richer } & \text { Thinner } & \text { Smarter } & \text { Younger } & \begin{array}{l}\text { None of } \\\\\text { these }\end{array} & \text { Total } \\\\\hline \text { Male } & 520 & 158 & 159 & 181 & 102 & 1120 \\\\\hline \text { Female } & 425 & 300 & 144 & 81 & 92 & 1042 \\\\\hline \text { Total } & 945 & 458 & 303 & 262 & 194 & 2162\end{array}$$ (a) How many adult Americans were surveyed? How many males were surveyed? (b) Construct a relative frequency marginal distribution. (c) What proportion of adult Americans want to be richer? (d) Construct a conditional distribution of desired trait by gender. That is, construct a conditional distribution treating gender as the explanatory variable. (e) Draw a bar graph of the conditional distribution found in part (d). (f) Write a couple sentences explaining any relation between desired trait and gender.

Step by step solution

01

- Total Number of Adult Americans Surveyed

Sum all the individuals in the 'Total' row or column of the contingency table. Total American Adults Surveyed = 2162

02

- Total Number of Males Surveyed

Sum all the individuals in the 'Male' row of the contingency table. Total Males Surveyed = 1120

03

- Construct a Relative Frequency Marginal Distribution

To create the relative frequency marginal distribution, divide each marginal total by the grand total.Relative frequencies:- Richer: \(\frac{945}{2162} ≈ 0.437\)- Thinner: \(\frac{458}{2162} ≈ 0.212\)- Smarter: \(\frac{303}{2162} ≈ 0.140\)- Younger: \(\frac{262}{2162} ≈ 0.121\)- None of these: \(\frac{194}{2162} ≈ 0.090\)

04

- Proportion of Adult Americans Who Want to Be Richer

Divide the number of adults who want to be richer by the total number of adults surveyed.Proportion: \(\frac{945}{2162} ≈ 0.437\)

05

- Construct a Conditional Distribution Treating Gender as the Explanatory Variable

Divide each number in the 'Male' and 'Female' rows by their respective row totals (1120 for males, 1042 for females).Conditional Distributions:- Males: Richer: \(\frac{520}{1120} ≈ 0.464\) Thinner: \(\frac{158}{1120} ≈ 0.141\) Smarter: \(\frac{159}{1120} ≈ 0.142\) Younger: \(\frac{181}{1120} ≈ 0.162\) None of these: \(\frac{102}{1120} ≈ 0.091\)- Females: Richer: \(\frac{425}{1042} ≈ 0.408\) Thinner: \(\frac{300}{1042} ≈ 0.288\) Smarter: \(\frac{144}{1042} ≈ 0.138\) Younger: \(\frac{81}{1042} ≈ 0.078\) None of these: \(\frac{92}{1042} ≈ 0.088\)

06

- Draw a Bar Graph of the Conditional Distribution

Create a bar graph for the conditional distribution found in Step 5. Use different colors or patterns for males and females and label the bars accordingly. The x-axis can represent the categories: Richer, Thinner, Smarter, Younger, None of these, and the y-axis should represent the proportions.

07

- Explain Any Relation Between Desired Trait and Gender

From the bar graph and conditional distribution, observe if there are notable trends. For example, males have a higher preference for being richer compared to females, while females have a higher preference for being thinner than males.

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Key Concepts

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

Random Sample

A random sample is a fundamental concept in statistics. It refers to a subset of individuals chosen from a larger population, where each individual has an equal chance of being selected. This method helps ensure that the sample represents the population as accurately as possible.
In this exercise, adult Americans were randomly sampled to determine their preference for certain desired traits like being richer, thinner, smarter, younger, or none of these.
Random sampling is key to producing valid and unbiased results. By including a diverse range of individuals, the survey findings reflect the general sentiments of the entire adult American population.
Here are some benefits of using a random sample:

• Reduces bias
• Enhances the accuracy of statistical predictions
• Ensures diversity in the sample

This method is widely used in various fields such as market research, medical studies, and social sciences.

Relative Frequency Distribution

Relative frequency distribution helps us understand how often a particular event occurs relative to the total number of observations.
Instead of just counting how many times each outcome occurs, relative frequency shows these counts as proportions or percentages. This way, we can compare different categories directly and observe trends more effectively.
In our exercise, this involves calculating the proportion of survey participants who chose each of the desired traits (Richer, Thinner, Smarter, Younger, None of these).
Steps to construct a relative frequency distribution:

• Determine the total number of observations (e.g., total survey participants).
• For each category (e.g., Richer, Thinner), divide the number of occurrences by the total number of observations.
• Express the result as a proportion or percentage.

For example, to find the relative frequency of participants who wanted to be richer, we calculate \ \(\( \frac{945}{2162} \approx 0.437 \)\).
Relative frequency distribution is especially useful in making comparisons and identifying trends within data sets.

Conditional Distribution

Conditional distribution shows the probability of one event occurring given that another event has already occurred. It helps in understanding the relationship between two or more variables.
In our exercise, treating gender as the explanatory variable allows us to see how preferences for desired traits differ between males and females.
To construct a conditional distribution:

• Select the explanatory variable (e.g., gender).
• Calculate the proportion for each category within the levels of the explanatory variable.

For instance, to find out the proportion of males who want to be richer, we use \ \( \( \frac{520}{1120} \approx 0.464 \) \ \). Similarly, for females wanting to be thinner, the proportion is \ \( \( \frac{300}{1042} \approx 0.288 \) \ \).
This method clearly shows the differences and similarities in preferences between genders and helps in drawing more nuanced conclusions.