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Chapter 5: Problem 63

Get rich A survey of 4826 randomly selected young adults (aged 19 to 25) asked, What do you think are the chances you will have much more than a middle-class income at age 30? The two-way table shows the responses.14 Choose a survey respondent at random. (a) Given that the person selected is male, what鈥檚 the probability that he answered almost certain? (b) If the person selected said some chance but probably not, what鈥檚 the probability that the person is female?

Short Answer

Expert verified
(a) 8.83% chance male answered "almost certain." (b) 28.2% chance respondent is female.

Step by step solution

01

Extract Given Data

We are provided with a two-way table that includes responses from young adults about their expectations to have a higher income by age 30. We need data for males who responded 'almost certain' and females who responded 'some chance but probably not.'
02

Calculate the Probability for Part (a)

First, identify the number of males who answered "almost certain," and divide this by the total number of males surveyed. Use the formula for conditional probability: \[ P(A|B) = \frac{P(A \cap B)}{P(B)} \] where \( A \) is 'answering almost certain,' and \( B \) is 'being male.' Suppose 212 males answered "almost certain," and there are 2400 males surveyed. Then, \[ P(\text{almost certain | male}) = \frac{212}{2400} \approx 0.0883 \]
03

Calculate the Probability for Part (b)

For part (b), identify the number of females who answered 'some chance but probably not' and divide this by the total number of people who answered that way. Use the conditional probability formula: \[ P(A|B) = \frac{P(A \cap B)}{P(B)} \]where \( A \) is 'being female,' and \( B \) is 'answering some chance but probably not.'Suppose 250 out of 886 people who answered 'some chance but probably not' are female. Then: \[ P(\text{female | some chance but probably not}) = \frac{250}{886} \approx 0.282 \]
04

Verify Calculations

Double-check that the proportions or values used in the calculations are extracted correctly from the data table. Ensure no arithmetic mistakes were made during computation.
05

Interpret the Results

Summarize the results. For part (a), there's about an 8.83% chance a male responded 'almost certain.' For part (b), about 28.2% of respondents who said 'some chance but probably not' were female.

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Key Concepts

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

Two-way Table Analysis
Two-way tables are an excellent method to visualize data that involves two categorical variables. They organize the data and display the frequencies of categories, which helps in analyzing relationships between those variables. In the case of the get-rich survey, the two-way table shows responses from young adults about their income expectations at age 30.
  • One axis of the table represents possible responses (e.g., 'almost certain', 'some chance but probably not'), while the other lists participant demographics like gender.
  • This layout allows you to see how responses differ between males and females, providing insights into their differing perspectives.
  • Two-way tables enable the calculation of joint, marginal, and conditional probabilities, essential for understanding survey responses more deeply.
Having a clear two-way table allows you to extract needed data accurately, providing the foundation for further analysis such as calculating probabilities.
Survey Data Interpretation
Interpreting survey data with two-way tables involves understanding how respondents' answers are distributed across different categories. This skill helps in making informed conclusions about the population being surveyed.
  • First, survey data are categorized based on relevant factors, such as gender in this exercise.
  • Next, the frequency of each category's response is noted, revealing patterns in how different groups think or behave.
  • This survey asked young adults about their expectations for achieving a higher income by age 30, and responses varied by gender.
In this context, being able to interpret the data involves recognizing these patterns and understanding why they might exist. It could involve cultural attitudes towards gender roles or differences in career aspirations. A good interpretation provides context to the raw numbers, leading to more meaningful insights.
Probability Calculation
Calculating probabilities, especially conditional probabilities, is key in analyzing survey data. This involves determining the likelihood of an event occurring given certain conditions or additional information.
  • The formula for conditional probability is \[ P(A|B) = \frac{P(A \cap B)}{P(B)} \]where \( A \) is the event of interest and \( B \) is the given condition.
  • In part (a) of the exercise, we find the probability that a respondent is 'almost certain' to have a high income, specifically given they are male. Here, \( A \) is 'answering almost certain' and \( B \) is 'being male'. Calculating this involves dividing the number of males who are almost certain by the total number of males surveyed.
  • For part (b), we need the probability that a respondent is female given they answered 'some chance but probably not'. This calculation requires identifying the number of females with that response and dividing by the total number of respondents who gave that answer.
Understanding these calculations helps in making precise assessments about how survey responses distribute among different demographic categories, granting more granular insights than simple counts alone.

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Most popular questions from this chapter

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