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Chapter 12: Problem 34

Education among young adults. Choose a young adult (aged 25-29) at random. The probability is \(0.10\) that the person chosen did not complete high school, \(0.27\) that the person has a high school diploma but no further education, and \(0.34\) that the person has at least a bachelor's degree. (a) What must be the probability that a randomly chosen young adult has some education beyond high school but does not have a bachelor's degree? (b) What is the probability that a randomly chosen young adult has at least a high school education?

Short Answer

Expert verified
(a) 0.29, (b) 0.90.

Step by step solution

01

Identify the Total Probability

The exercise provides the probabilities for different education levels among young adults aged 25-29. Let's identify these given probabilities: \( P(\text{No high school}) = 0.10 \), \( P(\text{Only high school}) = 0.27 \), and \( P(\text{At least bachelor's degree}) = 0.34 \). The total probability for any possible outcome must be 1.
02

Calculate Probability for Part (a)

We need to find the probability that a randomly chosen young adult has some education beyond high school but does not have a bachelor's degree. This is the probability of having some college or an associate's degree, calculated as:\[P(\text{Some college or associate's degree}) = 1 - P(\text{No high school}) - P(\text{Only high school}) - P(\text{At least bachelor's degree}).\]Substitute the known probabilities: \[P(\text{Some college or associate's degree}) = 1 - 0.10 - 0.27 - 0.34 = 0.29.\]
03

Calculate Probability for Part (b)

To find the probability that a young adult has at least a high school education, we consider those with a high school diploma and beyond. This includes the probabilities of having only high school, some college, or higher. Calculate as:\[P(\text{At least high school}) = 1 - P(\text{No high school}) = 1 - 0.10 = 0.90.\]

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Key Concepts

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

Educational attainment
When we talk about educational attainment, we're referring to the highest level of education that an individual has completed. This is an important demographic factor because it affects aspects such as employability and income levels. In the context of the problem at hand, we are looking at educational attainment among young adults aged 25-29. The data is categorized into distinct educational outcomes:
  • No high school diploma
  • High school diploma, but no further education
  • Bachelor's degree or higher
Each category represents a step farther in the educational journey, impacting opportunities and earning potential. By analyzing such data, researchers gain insights into education trends and socio-economic conditions.
Statistical analysis
Statistical analysis is a method of analyzing numerical data in order to draw conclusions. In our exercise, statistical analysis helps us understand educational attainment among young adults by assigning probabilities to various educational outcomes.

By knowing the probabilities of each educational category, we can perform calculations to determine additional probabilities. Without such statistical measures, it would be difficult to gauge the educational landscape effectively.

Distilling the given data into understandable probabilities
  • Measures the frequency of each educational outcome.
  • Allows prediction of future trends.
  • Enables comparisons across different groups.
Understanding these probabilities forms the basis of probability calculation, which is vital for making educated predictions about a population.
Probability calculation
Probability calculation is the statistical process of determining the likelihood of a given outcome. It forms the backbone of our solution strategy.
  • To find the probability of having some college or an associate's degree, we calculated it by subtracting the known probabilities from 1, the total probability of all possible educational outcomes.
  • The formula used: \[ P(\text{Some college or associate's degree}) = 1 - P(\text{No high school}) - P(\text{Only high school}) - P(\text{At least bachelor's degree}) \]
  • Similarly, for the probability of at least a high school education, we subtract the probability of not having a high school diploma, calculated as: \[ P(\text{At least high school}) = 1 - P(\text{No high school}) \]
By working through these calculations, we gain insight into the importance of probability in educational data. This provides a clear measure of different educational levels within a population, aiding in decisions and predictions.

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Most popular questions from this chapter

A basketball player shoots six free throws during a game. The sample space for counting the number she makes is (a) \(S=\) any number between 0 and 1 . (b) \(S=\) whole numbers 0 to 6 . (c) \(S=\) all sequences of six hits or misses, like HMMHHH.

Four-of-a-Kind. You read online that the probability of being dealt four-of- akind in a five-card poker hand is 1/4165. Explain carefully what this means. In particular, explain why it does not mean that if you are dealt 4165 five- card poker hands, one will be four-of-a-kind.

Survey accuracy. A sample survey contacted an SRS of 2854 registered voters shortly before the 2012 presidential election and asked respondents whom they planned to vote for. Election results show that \(51 \%\) of registered voters voted for Barack Obama. We will see later that in this situation the proportion of the sample who planned to vote for Barack Obama (call this proportion \(V\) ) has approximately the Normal distribution with mean \(\mu=0.51\) and standard deviation \(\sigma=0.009\). (a) If the respondents answer truthfully, what is \(P(0.49 \leq V \leq 0.53)\) ? This is the probability that the sample proportion \(V\) estimates the population proportion \(0.51\) within plus or minus \(0.02\). (b) In fact, \(49 \%\) of the respondents said they planned to vote for Barack Obama \((V=0.49)\). If respondents answer truthfully, what is \(P(V \leq 0.49)\) ?

A taste test. A tea-drinking Canadian friend of yours claims to have a very refined palate. She tells you that she can tell if, in preparing a cup of tea, milk is first added to the cup and then hot tea poured into the cup, or the hot tea is first poured into the cup and then the milk is added. \({ }^{18}\) To test her claims, you prepare six cups of tea. Three have the milk added first and the other three the tea first. In a blind taste test, your friend tastes all six cups and is asked to identify the three that had the milk added first. (a) How many different ways are there to select three of the six cups? (Hint: See Example 12.8.) (b) If your friend is just guessing, what is the probability that she correctly identifies the three cups with the milk added first?

Nickels falling over. You may feel that it is obvious that the probability of a head in tossing a coin is about \(1 / 2\) because the coin has two faces. Such opinions are not always correct. Stand a nickel on edge on a hard, flat surface. Pound the surface with your hand so that the nickel falls over. What is the probability that it falls with heads upward? Make at least 50 trials to estimate the probability of a head.
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