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

102030405060708090100

City A mean:

69.62

thousand dollarsCity A standard deviation:

22.26

thousand dollarsCity B

25

20

15

10

5

0

102030405060708090100

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.

To help the committee, let's start by analyzing which data set is appropriate for using the Empirical Rule.

### (a) Identifying the Appropriate Data Set:
The **Empirical Rule** (or 68-95-99.7 rule) is applicable when the data follows a normal distribution, which is bell-shaped and symmetric.

Given that the histograms are described in the problem, we can infer:
- **City A** has a higher standard deviation, and this might suggest a more spread out distribution.
- **City B** has a lower standard deviation and likely a more symmetric shape.

Assuming the histograms, we would generally expect that City B's data set would be more suitable for the Empirical Rule if its distribution is more symmetric and bell-shaped. However, without specific details about the histograms, the mean and standard deviation suggest that **City B** has a more narrowly distributed and possibly more normal-like distribution.

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

### (b) Approximation Using the Empirical Rule:
The Empirical Rule states:
- **68%** of data falls within 1 standard deviation of the mean.
- **95%** of data falls within 2 standard deviations of the mean.
- **99.7%** of data falls within 3 standard deviations of the mean.

For City B:
- Mean = 54.74 thousand dollars
- Standard Deviation = 13.77 thousand dollars

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:
- \( 13.43 = 54.74 - 3 \times 13.77 \) (3 standard deviations below the mean)
- \( 96.05 = 54.74 + 3 \times 13.77 \) (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:
68% of the data lies within 1 standard deviation of the mean.

Calculate the range:
- 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. What would be the interval for 95% of incomes in City B using the Empirical Rule?
2. How does the standard deviation impact the spread of data in a normal distribution?
3. Can the Empirical Rule be used for data sets with significant skewness? Why or why not?
4. What are some alternative methods to approximate data percentages if the Empirical Rule is not applicable?
5. How could the committee assess the normality of the income distributions visually?

**Tip:** When analyzing data, always consider the shape of the distribution before applying rules like the Empirical Rule to ensure accuracy.

Explore how the Empirical Rule applies to income distributions in City A and City B. Analyze the mean, standard deviation, and percentage of incomes falling within specific ranges. City B's income distribution is examined using statistical approximations, providing insights for urban planning committees.

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