The Higher Education Research Institute at UCLA collected data from 203,967 incoming first-time, full-time freshmen from 270 four-year colleges and universities in the U.S. 71.9% of those students replied that, yes, they believe that same-sex couples should have the right to legal marital status. Suppose that you randomly pick twelve first-time, full-time freshmen from the survey. You are interested in the number that believes that same-sex couples should have the right to legal marital status.

What is the standard deviation (𝜎)? (Round your answer to four decimal places.)
𝜎 =

To calculate the standard deviation for the number of students who believe that same-sex couples should have the right to legal marital status, we are dealing with a **binomial distribution**. The standard deviation of a binomial distribution is given by the formula:

\[
\sigma = \sqrt{n \cdot p \cdot (1 - p)}
\]

Where:
- \(n\) is the number of trials (in this case, the number of students picked),
- \(p\) is the probability of success (the proportion of students who believe in same-sex marriage),
- \(1 - p\) is the probability of failure (the proportion of students who do not believe in same-sex marriage).

### Given:
- \(n = 12\) (the number of students picked),
- \(p = 0.719\) (the proportion of students who believe in same-sex marriage),
- \(1 - p = 0.281\) (the proportion of students who do not believe in same-sex marriage).

### Now, calculate the standard deviation:

\[
\sigma = \sqrt{12 \cdot 0.719 \cdot 0.281}
\]

Let's compute that:

\[
\sigma = \sqrt{12 \cdot 0.719 \cdot 0.281} \approx \sqrt{2.426} \approx 1.558
\]

So, the standard deviation is approximately:

\[
\sigma \approx 1.5580
\]

### Final answer:
\[
\sigma = 1.5580
\]

Would you like further clarification or additional examples related to this problem? Here are some related questions you might find helpful:

1. How would the standard deviation change if the sample size was increased to 30 students?
2. What would happen to the standard deviation if the proportion \(p\) was lower, say 0.5?
3. How do you calculate the mean of a binomial distribution?
4. What is the probability that exactly 7 out of 12 students believe in same-sex marriage?
5. How would you apply this concept to a non-binomial distribution?

Tip: For binomial distributions, the mean is given by \(\mu = n \cdot p\).

The Higher Education Research Institute at UCLA collected data from 203,967 incoming first-time, full-time freshmen from 270 four-year colleges and universities in the U.S. 71.9% of those students replied that, yes, they believe that same-sex couples should have the right to legal marital status. Suppose that you randomly pick twelve first-time, full-time freshmen from the survey. You are interested in the number that believes that same-sex couples should have the right to legal marital status. What is the standard deviation (𝜎)? (Round your answer to four decimal places.) 𝜎 =

Learn how to calculate the standard deviation for a binomial distribution with a real-world example. This problem involves finding the standard deviation for a sample of 12 students from a larger survey, where 71.9% believe in same-sex marriage. Detailed solution provided for Grades 9-12 students.

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