A County Board of Supervisors has appointed an urban planning committee to evaluate proposed community development projects. The committee is analyzing, among other things, data on household incomes in two cities within the county. They have collected data on the income of 78

 households in each of the two cities. The histograms below show the distributions of the two sets of incomes (in thousands of dollars). Each histogram shows household income on the horizontal axis and number of households on the vertical axis. The means and standard deviations for the data sets are also given.City A

25

20

15

10

5

0




City A mean: 

69.62

 thousand dollarsCity A standard deviation: 

22.26

 thousand dollarsCity B

25

20

15

10

5

0




City B mean: 

54.74

 thousand dollarsCity B standard deviation: 

13.77

 thousand dollars

The committee wants to use the Empirical Rule to make some approximations about both data sets. Unfortunately, it is appropriate to use the Empirical Rule on only one of them!

Answer the parts below to help the committee with their approximations.

(a)Identify the data set for which it is appropriate to use the Empirical Rule.It is appropriate to use the Empirical Rule for the ▼

(Choose one)

 data set.

For the data set identified in part (a), use the Empirical Rule to make the following approximations.

(b)The percentage of incomes between 13.43

 thousand dollars and 96.05

 thousand dollars is approximately ▼

(Choose one)

.

(c)Approximately 68

%

 of the incomes are between 

 thousand dollars and 

 thousand dollars.

Let's break down the problem step by step to help the committee make the necessary approximations using the Empirical Rule.

### (a) Identifying the Appropriate Data Set:
The **Empirical Rule** is applicable for data that follows a normal distribution, which is characterized by a bell-shaped, symmetric curve.

Given the information:
- **City A** has a mean of 69.62 thousand dollars and a standard deviation of 22.26 thousand dollars.
- **City B** has a mean of 54.74 thousand dollars and a standard deviation of 13.77 thousand dollars.

Since we don't have the actual histograms, we can infer that City B, with a smaller standard deviation, might have a distribution that is more symmetric and closer to a normal distribution compared to City A, which has a larger spread.

Thus, **it is appropriate to use the Empirical Rule for the City B data set**.

### (b) Approximation Using the Empirical Rule:
Using the Empirical Rule for City B, we know:
- Mean = 54.74 thousand dollars
- Standard Deviation = 13.77 thousand dollars

We want to find the percentage of incomes between **13.43** thousand dollars and **96.05** thousand dollars.

First, calculate how many standard deviations these values are from the mean:
- Lower Bound: \( 13.43 = 54.74 - 3 \times 13.77 \) (3 standard deviations below the mean)
- Upper Bound: \( 96.05 = 54.74 + 3 \times 13.77 \) (3 standard deviations above the mean)

According to the Empirical Rule:
- **99.7%** of the data falls within 3 standard deviations of the mean.

So, **the percentage of incomes between 13.43 thousand dollars and 96.05 thousand dollars is approximately 99.7%**.

### (c) Approximating the 68% Interval:
The Empirical Rule states that 68% of the data lies within 1 standard deviation of the mean.

For City B:
- Lower Bound: \( 54.74 - 13.77 = 40.97 \) thousand dollars
- Upper Bound: \( 54.74 + 13.77 = 68.51 \) thousand dollars

So, **approximately 68% of the incomes are between 40.97 thousand dollars and 68.51 thousand dollars**.

---

Would you like more details or have any questions? Here are five related questions:

1. How would the interval change if we were interested in the 95% range for City B?
2. What might cause a data set to not be appropriate for the Empirical Rule?
3. How could the committee use a histogram to visually assess the normality of a data set?
4. What are some examples of distributions where the Empirical Rule would not apply?
5. How can understanding income distribution help in making decisions about community development?

**Tip:** Always check the shape of the data distribution before applying the Empirical Rule, as it only applies to normal or approximately normal distributions.

Learn how to use the Empirical Rule to approximate income distributions in City B, analyzing data on household incomes with mean 54.74 thousand dollars and standard deviation 13.77 thousand dollars. Understand the percentage of incomes between 13.43 thousand dollars and 96.05 thousand dollars and the 68% interval. Suitable for advanced high school students studying statistics and normal distribution concepts.

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