According to a college survey, 22% of all students work full time. Find the mean for the random variable X, the number of students who work full time in samples of size 16.

To find the mean for the random variable \( X \), representing the number of students who work full-time in samples of size 16, we can use the formula for the mean of a binomial distribution:

\[
\text{Mean} = n \cdot p
\]

where:
- \( n \) is the sample size (number of trials),
- \( p \) is the probability of success (the probability that a student works full time).

Given:
- \( n = 16 \)
- \( p = 0.22 \)

### Calculation

\[
\text{Mean} = 16 \times 0.22 = 3.52
\]

Thus, the mean number of students who work full time in samples of size 16 is **3.52**.

Would you like further details on this, or do you have any questions?

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Here are some related questions to explore:

1. What is the standard deviation for this binomial distribution?
2. How would the mean change if the probability of working full time increased to 30%?
3. What is the probability that exactly 4 students work full time in a sample of 16?
4. How does the sample size affect the mean and standard deviation in binomial distributions?
5. Can this situation be approximated by a normal distribution, and if so, how?

**Tip:** In binomial distributions, the mean depends on both the sample size and the probability of success.

This problem demonstrates how to calculate the mean of a binomial distribution where 22% of students work full time. Using a sample size of 16 and applying the formula Mean = n * p, we find the mean number of students working full time. Suitable for high school students in Grades 10-12.

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