A study found that 1% of Social Security recipients are too young to vote. If 800 Social Security recipients are randomly selected, find the mean, variance, and standard deviation of the number of recipients who are too young to vote.

This problem is an example of a **binomial distribution**. In a binomial distribution, there are two outcomes: success (a recipient is too young to vote) or failure (a recipient is not too young to vote).

Given:
- Probability of "success" (recipient is too young to vote) = \( p = 0.01 \)
- Probability of "failure" (recipient is not too young to vote) = \( q = 1 - p = 0.99 \)
- Number of trials (recipients selected) = \( n = 800 \)

For a binomial distribution, we can calculate the mean (\( \mu \)), variance (\( \sigma^2 \)), and standard deviation (\( \sigma \)) using the following formulas:

### 1. Mean (\( \mu \)):
\[
\mu = n \times p
\]
\[
\mu = 800 \times 0.01 = 8
\]

### 2. Variance (\( \sigma^2 \)):
\[
\sigma^2 = n \times p \times q
\]
\[
\sigma^2 = 800 \times 0.01 \times 0.99 = 7.92
\]

### 3. Standard deviation (\( \sigma \)):
\[
\sigma = \sqrt{\sigma^2}
\]
\[
\sigma = \sqrt{7.92} \approx 2.81
\]

### Summary:
- Mean (\( \mu \)): 8
- Variance (\( \sigma^2 \)): 7.92
- Standard deviation (\( \sigma \)): 2.81

Do you want more details on how these formulas are derived or used? Feel free to ask any follow-up questions!

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Here are 5 related questions you can explore:
1. How would the results change if the percentage of recipients too young to vote increased to 5%?
2. What is the probability that exactly 8 recipients are too young to vote?
3. What is the probability that fewer than 5 recipients are too young to vote?
4. How does increasing the sample size to 1,000 affect the mean and standard deviation?
5. How does the binomial distribution approximate the normal distribution for large sample sizes?

**Tip:** In large sample sizes, binomial distributions can often be approximated by normal distributions for easier calculations!

Learn how to calculate the mean, variance, and standard deviation for the number of Social Security recipients too young to vote using binomial distribution. This problem covers key statistical concepts and is suitable for students in Grades 10-12.

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