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The unemployment rate is a measure of the prevalence of unemployment within an economy. It indicates the percentage of employable people in a labor force who are not working. In this context, the term "employable" refers to people who are of working age and actively seeking employment. For Arizona, the unemployment rate was given as 4.1%. This means for every 100 employable people, roughly 4 are out of work.
This statistic is essential in understanding labor market health. A low unemployment rate suggests that the labor market is performing well, with most people who want to work being able to find jobs. Conversely, a high unemployment rate may signify economic troubles as more individuals are unable to secure employment.
By understanding this percentage, policymakers and economists can gauge economic stability and deploy resources or strategies to boost employment opportunities.

The expected value in a statistical context is essentially a measure of the center of a probability distribution. It's like the average you would expect to see in the long run for your data. For a binomial distribution—which fits perfectly when considering the outcome of a finite number of binary events like 'employed or not employed'—the expected value formula is:

• \( E(X) = n \times p \)

where:

• \( n \) is the number of trials or individuals, in our case, 100 people.
• \( p \) is the probability of success, which here is the probability of someone being unemployed, given as 0.041.

Plugging in the numbers, you find that \( E(X) = 100 \times 0.041 = 4.1 \).
This means, out of 100 employable people, we expect approximately 4 to be unemployed, aligning with the 4.1% unemployment rate.

Variance is a measure of how spread out the values in a distribution are. In simple terms, it tells us how much the data deviate from the expected value, on average. For a binomial distribution, the variance formula is:

• \( Var(X) = n \times p \times (1-p) \)

Here:

• \( n = 100 \)
• \( p = 0.041 \)

Filling it out, \( Var(X) = 100 \times 0.041 \times 0.959 = 3.9359 \).
This variance indicates the degree of variability or dispersion from the expected unemployment number, which is 4.1. A lower variance would imply that most people's unemployment statuses closely align with this expected value, while a higher variance would suggest more variability.

Standard deviation is a statistical measure that expresses the amount of variation or dispersion in a set of values. It is a crucial concept because it gives a more interpretable scale compared to variance. While variance is in squared units, standard deviation returns it to original units.
The formula for standard deviation is the square root of the variance:

• \( \sigma(X) = \sqrt{Var(X)} \)

So, \( \sigma(X) = \sqrt{3.9359} \approx 1.982 \).
This means the typical deviation from the average number of unemployed people, 4.1, is approximately 1.982 persons. This statistic lets us understand, in more practical terms, the typical range we can expect our sample's unemployment number to vary from the expected value of 4.1.