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Chapter 5: Problem 59

The unemployment rate in the state of Arizona is \(4.1 \%\) (CNN Money website, May 2 , 2007 ). Assume that 100 employable people in Arizona are selected randomly. a. What is the expected number of people who are unemployed? b. What are the variance and standard deviation of the number of people who are unemployed?

Short Answer

Expert verified
Expected number unemployed: 4.1, Variance: 3.9359, Standard deviation: 1.984.

Step by step solution

01

Understand the Problem

We need to determine the expected number of unemployed people and the variance and standard deviation of this number when randomly selecting 100 employable people in Arizona, given the unemployment rate is 4.1%.
02

Expectation Calculation

The expected number of unemployed people is calculated using the formula for the expectation of a binomial distribution. The expectation (mean) is given by \[ E(X) = n \times p \],where \( n = 100 \) (the number of trials) and \( p = 0.041 \) (the probability of unemployment). So, \[ E(X) = 100 \times 0.041 = 4.1. \]
03

Calculate Variance

Variance for a binomial distribution is calculated using the formula \[ Var(X) = n \times p \times (1-p). \]Substituting the values, we have \[ Var(X) = 100 \times 0.041 \times (1 - 0.041) = 100 \times 0.041 \times 0.959 = 3.9359. \]
04

Calculate Standard Deviation

The standard deviation is the square root of the variance. Hence, \[ \sigma = \sqrt{Var(X)} = \sqrt{3.9359} \approx 1.984. \]

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Key Concepts

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

Binomial Distribution
In the world of probability, the binomial distribution is a powerful tool that helps us model scenarios with two possible outcomes, like success or failure. To put it simply, it is especially useful in analyzing situations where we have a fixed number of independent trials, each with the same probability of a particular outcome. This is what makes it suitable for calculating probabilities of unemployment, as demonstrated in the exercise.

For a binomial distribution, we have:
  • "n" trials (e.g., the 100 selected people in Arizona).
  • Probability "p" for success in each trial (0.041 probability of being unemployed).
  • Each trial is independent, meaning the outcome of one doesn鈥檛 affect the others.
Recognizing these characteristics makes it easier to calculate properties like expectation, variance, and standard deviation, which are pivotal for interpreting the results.
Expectation
The expectation or mean of a binomial distribution provides the average we expect from the distribution over several iterations, based on the given probability. In the context of unemployment, it gives us the expected number of unemployed people.

For a binomial distribution, the formula to calculate expectation is:
  • \[ E(X) = n imes p \]
  • "n" is the number of trials, and "p" is the probability of unemployment here.
In our exercise, calculating the expected number of unemployed people among 100 individuals at a 4.1% unemployment rate, we find:
  • \[ E(X) = 100 imes 0.041 = 4.1 \]
In simpler terms, on average, we expect to find 4.1 unemployed people out of the 100 randomly selected.
Variance
Variance helps us understand how much the values can differ from the expected value in a binomial distribution. For our unemployment example, it tells us the spread around the expected number of unemployed people.

The formula for variance in a binomial distribution is:
  • \[ Var(X) = n imes p imes (1-p) \]
  • "n" is the number of trials, "p" is the probability of unemployment, and "1-p" represents the probability of being employed.
For our exercise:
  • \[ Var(X) = 100 imes 0.041 imes (1 - 0.041) \]
  • \[ = 100 imes 0.041 imes 0.959 = 3.9359 \]
This variance tells us how spread out the number of unemployed people might be from our expected 4.1.
Standard Deviation
The standard deviation is a measure that simply provides insight into the average distance each data point is from the mean. For our binomial distribution, it鈥檚 derived by taking the square root of the variance. This can help us understand the typical variation in the number of unemployed people compared to our expectation.

Using our variance, we calculate the standard deviation as:
  • \[ \sigma = \sqrt{Var(X)} \]
  • \[ = \sqrt{3.9359} \approx 1.984 \]
This value, approximately 1.984, tells us that typically, the number of unemployed people deviates by about 1.984 people from the expected 4.1 people, offering insights into the variability of our data.

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