An agency collects demographics concerning the number of people in families per household in a certain country. Assume the distribution of the number of people per household is as shown in the table to the right.

a. nbspa. 

Calculate the expected number of people in families per household in the country.

b.

Compute the standard deviation of the number of people in families per household.

nbsp nbsp  

**x**

​P(x)

nbsp nbsp  
 
22

0.270.27

33

0.250.25

44

0.290.29

55

0.140.14

66

0.030.03

77

0.020.02

Question content area bottom

Part 1

a. The expected number of people in families per household is 

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Let's solve this step by step. 

### Part A: Expected Number of People per Household

The expected value \( E(X) \), or the mean, of the number of people per household can be found using the formula for the expected value of a discrete random variable:

\[
E(X) = \sum [x \cdot P(x)]
\]

Where:
- \( x \) is the number of people per household
- \( P(x) \) is the probability of having \( x \) people in a household

Given the dataLet's solve this step by step. 

### Part A: Expected Number of People per Household

The expected value \( E(X) \), or the mean, of the number of people per household can be found using the formula for the expected value of a discrete random variable:

\[
E(X) = \sum [x \cdot P(x)]
\]

Where:
- \( x \) is the number of people per household
- \( P(x) \) is the probability of having \( x \) people in a household

Given the 

\[
\begin{array}{|c|c|}
\hline
x & P(x) \\
\hline
2 & 0.27 \\
3 & 0.25 \\
4 & 0.29 \\
5 & 0.14 \\
6 & 0.03 \\
7 & 0.02 \\
\hline
\end{array}
\]

Now, multiply each \( x \) by its corresponding probability \( P(x) \), and sum the results:

\[
E(X) = (2 \times 0.27) + (3 \times 0.25) + (4 \times 0.29) + (5 \times 0.14) + (6 \times 0.03) + (7 \times 0.02)
\]

\[
E(X) = 0.54 + 0.75 + 1.16 + 0.7 + 0.18 + 0.14 = 3.47
\]

So, the expected number of people per household is **3.47**.

### Part B: Standard Deviation of the Number of People per Household

The standard deviation \( \sigma \) is calculated using the formula:

\[
\sigma = \sqrt{ E(X^2) - [E(X)]^2 }
\]

We already know \( E(X) = 3.47 \). Now, we need to calculate \( E(X^2) \), the expected value of the squares of \( X \), using the formula:

\[
E(X^2) = \sum [x^2 \cdot P(x)]
\]

Let's calculate \( x^2 \cdot P(x) \) for each value of \( x \):

\[
E(X^2) = (2^2 \times 0.27) + (3^2 \times 0.25) + (4^2 \times 0.29) + (5^2 \times 0.14) + (6^2 \times 0.03) + (7^2 \times 0.02)
\]

\[
E(X^2) = (4 \times 0.27) + (9 \times 0.25) + (16 \times 0.29) + (25 \times 0.14) + (36 \times 0.03) + (49 \times 0.02)
\]

\[
E(X^2) = 1.08 + 2.25 + 4.64 + 3.5 + 1.08 + 0.98 = 13.53
\]

Now, apply the formula for the standard deviation:

\[
\sigma = \sqrt{E(X^2) - [E(X)]^2} = \sqrt{13.53 - (3.47)^2}
\]

\[
\sigma = \sqrt{13.53 - 12.04} = \sqrt{1.49} = 1.22
\]

Thus, the standard deviation is approximately **1.22**.

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Do you have any questions or want more details about the process?

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Here are 5 related questions:
1. How does the expected value change if the probability distribution changes?
2. What is the significance of standard deviation in this context?
3. How would the variance be affected if we increased the probability for households of 4 people?
4. Can the expected number of people per household ever be greater than 7? Why or why not?
5. What is the difference between standard deviation and variance?

**Tip**: The expected value is like a "weighted average" that takes into account the likelihood of each outcome.

An agency collects demographics concerning the number of people in families per household in a certain country. Assume the distribution of the number of people per household is as shown in the table. a. Calculate the expected number of people in families per household in the country. b. Compute the standard deviation of the number of people in families per household.

Learn how to calculate the expected number of people in families per household and the standard deviation, using probability distributions. This problem involves understanding expected value and standard deviation calculations for a given probability distribution.

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