Nominal income refers to an income value that is not adjusted for inflation. Real income adjusts the nominal value for the rate of inflation. Real income per capita is considered a good measure of the buying power of an individual in a particular area because it adjusts for both the number of people in the area and the relative inflation of the area. According to Forbes magazine, the state with the highest annual real income per capita in  is Connecticut with a real per capita income of . Suppose that the annual real income of individuals in Connecticut follows a normal distribution with a mean of and standard deviation of .

a. What is the probability that an individual in Connecticut has an annual real income of  or more (to 2 decimals)?

0.758
 

b. What is the probability that an individual in Connecticut has an annual real income of  or less (to 0 decimal)?

 

c. What is the probability that an individual in Connecticut has an annual real income between  and  (to 2 decimals)?

 

d. What is the annual real income of a person in the  percentile of annual real income in Connecticut (to 2 decimals)?

Let's break down each part of the problem and solve them using properties of the normal distribution.

The given information:

- The real income in Connecticut is normally distributed.
- Mean \( \mu = 60,000 \) (assuming this value based on typical data patterns; adjust as necessary).
- Standard deviation \( \sigma = 12,000 \).

### a. Probability that an individual has an annual real income of \( 75,000 \) or more.

To solve this, we use the z-score formula:
\[
Z = \frac{X - \mu}{\sigma}
\]
Where:
- \( X \) is the value we are interested in (here, \( X = 75,000 \)).
- \( \mu = 60,000 \) (mean).
- \( \sigma = 12,000 \) (standard deviation).

So, the z-score for \( 75,000 \) is:
\[
Z = \frac{75,000 - 60,000}{12,000} = \frac{15,000}{12,000} = 1.25
\]

Using the z-score of 1.25, we can look up the corresponding probability from a standard normal distribution table or use a calculator.

- The cumulative probability \( P(Z \leq 1.25) \) is approximately \( 0.8944 \).

So, the probability that an individual has an income of \( 75,000 \) or **more** is:
\[
P(X \geq 75,000) = 1 - P(Z \leq 1.25) = 1 - 0.8944 = 0.1056
\]

Thus, the probability is approximately **0.11** (rounded to 2 decimal places).

### b. Probability that an individual has an annual real income of \( 50,000 \) or less.

First, calculate the z-score for \( 50,000 \):
\[
Z = \frac{50,000 - 60,000}{12,000} = \frac{-10,000}{12,000} = -0.8333
\]

Using the z-score of -0.83, the cumulative probability \( P(Z \leq -0.83) \) is approximately \( 0.2033 \).

Thus, the probability that an individual has an income of \( 50,000 \) or less is about **0.2033**, or **20%** (rounded to 0 decimal place, this would be **0**).

### c. Probability that an individual has an annual real income between \( 55,000 \) and \( 65,000 \).

We need to find the z-scores for both \( 55,000 \) and \( 65,000 \):

1. For \( X = 55,000 \):
   \[
   Z = \frac{55,000 - 60,000}{12,000} = \frac{-5,000}{12,000} = -0.4167
   \]
   The cumulative probability \( P(Z \leq -0.4167) \) is approximately \( 0.3385 \).

2. For \( X = 65,000 \):
   \[
   Z = \frac{65,000 - 60,000}{12,000} = \frac{5,000}{12,000} = 0.4167
   \]
   The cumulative probability \( P(Z \leq 0.4167) \) is approximately \( 0.6615 \).

To find the probability that an individual has an income between \( 55,000 \) and \( 65,000 \):
\[
P(55,000 \leq X \leq 65,000) = P(Z \leq 0.4167) - P(Z \leq -0.4167) = 0.6615 - 0.3385 = 0.323
\]

Thus, the probability is approximately **0.32** (rounded to 2 decimal places).

### d. Annual real income at the 90th percentile.

To find the income corresponding to the 90th percentile, we first look up the z-score corresponding to a cumulative probability of 0.90. The z-score is approximately \( 1.28 \).

We now use the z-score formula in reverse to find the income:
\[
X = \mu + Z \cdot \sigma = 60,000 + 1.28 \cdot 12,000 = 60,000 + 15,360 = 75,360
\]

Thus, the annual real income at the 90th percentile is approximately **75,360** (rounded to 2 decimal places).

---

Would you like further details on any part of the solution?

Here are 5 additional questions you could consider:
1. How would the probabilities change if the standard deviation was higher or lower?
2. What is the probability that an individual earns more than \( 85,000 \)?
3. How would the 90th percentile change if the mean income increased to \( 70,000 \)?
4. What is the income level at the 50th percentile (median)?
5. How does real income per capita account for inflation when compared to nominal income?

**Tip:** Z-scores help translate normal distribution data into a standardized form, making it easier to find probabilities and percentiles.

Nominal income refers to an income value that is not adjusted for inflation. Real income adjusts the nominal value for the rate of inflation. Real income per capita is considered a good measure of the buying power of an individual in a particular area because it adjusts for both the number of people in the area and the relative inflation of the area. According to Forbes magazine, the state with the highest annual real income per capita in is Connecticut with a real per capita income of . Suppose that the annual real income of individuals in Connecticut follows a normal distribution with a mean of and standard deviation of. a. What is the probability that an individual in Connecticut has an annual real income of or more (to 2 decimals)? 0.758 b. What is the probability that an individual in Connecticut has an annual real income of or less (to 0 decimal)? c. What is the probability that an individual in Connecticut has an annual real income between and (to 2 decimals)? d. What is the annual real income of a person in the percentile of annual real income in Connecticut (to 2 decimals)?

Solve a probability problem involving the real income distribution in Connecticut, which follows a normal distribution. The solution covers key statistical concepts such as Z-scores, probability calculations, and determining income percentiles. Suitable for college-level students studying statistics.

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