Question

In Exercise 6.28 , we suggested that the IRS assign auditing rates per state by randomly selecting 50 auditing percentages from a normal distribution with a mean equal to \(1.55 \%\) and a standard deviation of \(.45 \%\) a. What is the probability that a particular state would have more than \(2 \%\) of its tax returns audited? b. What is the expected value of \(x\), the number of states that will have more than \(2 \%\) of their income tax returns audited? c. Is it likely that as many as 15 of the 50 states will have more than \(2 \%\) of their income tax returns audited?

Short answer

Answer: The expected value can be calculated by multiplying the probability found in step 2 by the total number of states (50). To determine if 15 states having more than 2% audited is likely, compare it to the expected value calculated in step 3.

Step-by-step solution

Calculate Z-score

The first step is to calculate the Z-score for the value 2% (0.02). The Z-score is the number of standard deviations a value is from the mean. The formula to calculate the Z-score is: \(Z = \frac{x - \mu}{\sigma}\) where \(x\) is the value (0.02), \(\mu\) is the mean (0.0155), and \(\sigma\) is the standard deviation (0.0045). Using this formula, we find the Z-score for 2%: \(Z = \frac{0.02 - 0.0155}{0.0045}\)

Find the Probability for a Z-score

After calculating the Z-score, we can use a Z-table (standard normal distribution table) to find the probability corresponding to that Z-score. Since we are looking for the probability of having more than 2% of tax returns audited, we need to find the area to the right of the Z-score we just calculated. Using a Z-table, we find the probability for our calculated Z-score.

Calculate the Expected Value of States with More Than 2% Audited

The expected value of the number of states with more than 2% of their tax returns audited can be calculated by multiplying the probability found in the previous step by the total number of states (50). Expected Value = Probability × Total Number of States

Determine If 15 States Having More Than 2% Audited is Likely

In order to determine if as many as 15 states having more than 2% of their tax returns audited is likely, we need to find the probability of at least 15 states having an auditing rate above 2%. Since the expected value gives us the average number of states with auditing rates above 2%, we can compare this to the value of 15 to see if it is likely or not.

Key concepts

Z-score Calculation

Understanding the Z-score is crucial when dealing with statistical data, especially when it relates to a normal distribution. A Z-score, also known as a standard score, quantifies how many standard deviations an element is from the mean of the distribution. The formula for calculating a Z-score is given by:
\[ Z = \frac{x - \mu}{\sigma} \]
Where \(x\) represents the value in question, \(\mu\) is the mean of the distribution, and \(\sigma\) is the standard deviation. To put this in the context of the exercise, if the mean auditing rate is 1.55%, and the standard deviation is 0.45%, for a state with a 2% auditing rate, the Z-score would tell us how 'unusual' this rate is compared to what's typical (the mean). A higher Z-score indicates a value further away from the mean, which in this auditing scenario, would mean a higher chance of being audited.
By calculating the Z-score, we can then proceed to look up this value in a standard normal distribution table, also known as a Z-table, to find out how likely (or unlikely) it is for a state to have more than 2% of its tax returns audited.

Normal Distribution

The normal distribution, often depicted as a bell curve, is a foundational concept in statistics that describes how the values of a variable are distributed. Values close to the mean are more frequent in occurrence, hence the bulge in the center of the curve. As you move away from the mean, towards either tail, the frequency of those values decreases, which is depicted by the tapering ends of the bell curve.
The area under the curve corresponds to probabilities, with the total area summing up to 1 or 100%. The mean of the normal distribution divides the curve into two symmetrical halves.
When we refer to 'normal distribution' in the context of the auditing rates, we are assuming that most states will have auditing rates close to the mean of 1.55%, and only a few will have values significantly higher or lower (like 2% or more). When you find the probability related to a Z-score, you’re essentially looking up which percentile that Z-score corresponds to on this bell curve. The exercise required finding the probability of a state having an auditing rate exceeding 2%, which translates to finding the area to the right of the Z-score on the normal distribution curve.

Expected Value

Expected value is a key concept in probability and statistics, representing the average or mean of a random variable’s potential outcomes, weighted by their respective probabilities. It provides a measure of the center of the distribution of the variable. In simple terms, it's a prediction of the average result if the same situation were to happen repeatedly.
In the scenario given in the exercise, one would calculate the expected value to predict the average number of states that would have more than 2% of their tax returns audited. This is done by taking the probability of a single state having an audit rate above 2%, as determined from the Z-score and the normal distribution table, and multiplying it by the total number of states (50). This gives us the expected number of states falling into this higher audit category.
Mathematically, for a set of outcomes \(x_1, x_2, ..., x_n\) with probabilities \(p_1, p_2, ..., p_n\), the expected value \(E[X]\) is given by: \[ E[X] = x_1p_1 + x_2p_2 + ... + x_np_n \]
In our exercise, using the expected value helps us answer whether it’s likely for as many as 15 states to have audit rates over 2% by providing a baseline for comparison. If the expected number is much lower than 15, it would indicate such a scenario is less likely.