The problem involves a binomial distribution where 76% of U.S. adults think that political correctness is a problem in America. You randomly select six U.S. adults and ask if they think political correctness is an issue.

Let's break down the steps to find the requested values:

Binomial Distribution Formula:

• Mean $\mu = n \cdot p$
• Variance $\sigma^2 = n \cdot p \cdot (1 - p)$
• Standard deviation $\sigma = \sqrt{n \cdot p \cdot (1 - p)}$

Where:

• $n = 6$ (the number of adults sampled),
• $p = 0.76$ (probability that an adult thinks political correctness is a problem).

Now, let's calculate the values:

1. Mean (μ): $\mu = 6 \cdot 0.76 = 4.56$

2. Variance (σ²): $\sigma^2 = 6 \cdot 0.76 \cdot (1 - 0.76) = 6 \cdot 0.76 \cdot 0.24 = 1.09$

3. Standard deviation (σ): $\sigma = \sqrt{1.09} \approx 1.04$

Interpretation:

• Mean (μ): On average, you would expect around 4.6 adults out of 6 to think that political correctness is a problem.
• Variance (σ²): The variance is 1.09, which indicates how much the number of people who think political correctness is a problem could vary from sample to sample.
• Standard deviation (σ): The standard deviation of about 1.04 shows the expected spread of the number of adults who think political correctness is a problem from the average.

Final Question:

• Most samples of 6 adults would differ from the mean by no more than 1.0 (rounded to the nearest tenth).

Let me know if you'd like further clarification or additional help! Here are some related questions you might find interesting:

1. How do you calculate probabilities using the binomial distribution?
2. What happens to the variance and standard deviation as the sample size increases?
3. How can we find the probability that exactly 4 adults in a sample of 6 think political correctness is a problem?
4. How does a higher probability (like 0.76) impact the results of the binomial distribution?
5. What is the difference between variance and standard deviation in the context of the binomial distribution?

Tip: Always check if your probability $p$ is between 0 and 1 when working with a binomial distribution!