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. 71.6% 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 =

This is a binomial probability problem, where we are interested in the standard deviation of a binomial distribution. The binomial distribution has the following properties:

- \( n \): number of trials (in this case, 12 freshmen)
- \( p \): probability of success (in this case, the probability that a student believes same-sex couples should have the right to legal marital status, which is 0.716)
- \( q = 1 - p \): probability of failure (i.e., the complement of the success probability, \( q = 1 - 0.716 = 0.284 \))

The standard deviation \( \sigma \) for a binomial distribution is given by the formula:

\[
\sigma = \sqrt{n \cdot p \cdot q}
\]

### Step-by-step calculation:

1. \( n = 12 \)
2. \( p = 0.716 \)
3. \( q = 1 - 0.716 = 0.284 \)

Now, plug these values into the formula:

\[
\sigma = \sqrt{12 \cdot 0.716 \cdot 0.284}
\]

First, calculate the product inside the square root:

\[
12 \cdot 0.716 \cdot 0.284 = 2.4411
\]

Then, take the square root:

\[
\sigma = \sqrt{2.4411} = 1.5623
\]

Thus, the standard deviation \( \sigma \) is approximately **1.5623**.

Would you like any additional details or clarifications?

### 5 Related Questions:
1. How do you calculate the mean of a binomial distribution?
2. What is the probability that exactly 8 out of the 12 students support same-sex marriage rights?
3. How would the standard deviation change if the sample size were increased to 20 students?
4. What assumptions must hold true for a distribution to be modeled as binomial?
5. How can the normal approximation be used for a binomial distribution?

### Tip:
For binomial distributions with large \(n\) and probabilities close to 0.5, you can use the normal approximation for faster calculations.

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. 71.6% 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.)

Learn how to calculate the standard deviation of a binomial distribution using data from a UCLA survey. With 12 randomly selected students and a 71.6% probability that they support same-sex marriage rights, this problem involves applying the binomial standard deviation formula. Suitable for Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation