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Chapter 5: Problem 4

In November the U.S. unemployment rate was \(8.7 \%\) (U.S. Department of Labor website, January 10,2010 ). The Census Bureau includes nine states in the Northeast region. Assume that the random variable of interest is the number of Northeastern states with an unemployment rate in November that was less than \(8.7 \%\). What values may this random variable have?

Short Answer

Expert verified
The random variable can have values: 0, 1, 2, ..., 9.

Step by step solution

01

Understanding the Problem

We need to find the possible values of a random variable that represents the number of Northeastern states with unemployment rates below 8.7%. Since there are nine states in the Northeast region, the random variable represents the count of qualified states out of these nine.
02

Identifying the Range of Values

Since we are counting the number of states, the possible outcomes for this random variable range from 0 to 9. This is because all states could potentially have unemployment rates below, above, or some mix below and above 8.7%. Hence, every whole number from 0 to 9 is a possible value for this count.
03

Finalize the Possible Values

Conclude that the possible values of the random variable are 0, 1, 2, ..., 8, 9. This represents all possibilities from no states having rates below 8.7% to all nine states having rates below 8.7%.

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Key Concepts

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

Unemployment Rate
When we talk about the unemployment rate, we're looking at the percentage of the workforce that is unemployed and actively seeking jobs. It's a significant indicator of economic health.
  • An unemployment rate of 8.7% means that out of every 100 people available for work, approximately 9 are jobless and looking for work.
  • This figure is calculated by dividing the number of unemployed individuals by the total labor force, then multiplying by 100 to get a percentage.
Tracking changes in this rate over time helps economists and policymakers understand economic trends and make crucial decisions. In the context of the exercise, knowing the unemployment rate helps to evaluate how each state compares to the national average or specific benchmarks such as 8.7%.
Statistical Range
The concept of statistical range in this context is about understanding the possible values a random variable can take.
  • For example, if there are nine states and the random variable is the number of states with an unemployment rate below 8.7%, the range is from 0 to 9.
  • This means there could be zero states, which is the minimum, or all nine states having unemployment rates below 8.7%, which is the maximum.
In statistics, the range gives us the simplest form of variability measurement, indicating the span between the lowest and highest observations in a dataset. Here it helps to quantify the spectrum of possibilities for the number of states with a lower unemployment rate than 8.7%.
Probability Distribution
Probability distribution is a statistical function that describes all the possible values and likelihoods that a random variable can take within a given range.

For this exercise, the random variable is the number of Northeastern states with an unemployment rate less than 8.7%.
  • A probability distribution would assign a probability to each possible number of states from 0 to 9 having unemployment rates below 8.7%.
  • For instance, if it is likely that 5 states have rates below the benchmark, the probability distribution would show a higher probability for the number 5 compared to numbers far from 5.
Understanding probability distributions is key because it helps predict outcomes and make data-driven decisions based on the likelihood of different scenarios.

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Most popular questions from this chapter

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Suppose \(N=10\) and \(r=3 .\) Compute the hypergeometric probabilities for the following values of \(n\) and \(x\) a. \(\quad n=4, x=1\) b. \(n=2, x=2\) c. \(\quad n=2, x=0\) d. \(n=4, x=2\) e. \(n=4, x=4\)
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