Let's break down the problem step by step.

(a) Calculating the mean and standard deviation

You are dealing with a binomial distribution in this case, since you are selecting a sample of 50 adults and checking whether each individual has never been married. We can approximate this binomial distribution as a normal distribution using the central limit theorem because the sample size is large.

Given:

• Proportion of adults who have never been married: $p = 0.21$
• Sample size: $n = 50$

The mean and standard deviation for a binomial distribution are given by:

$\text{Mean} = \mu = n \cdot p$ $\text{Standard deviation} = \sigma = \sqrt{n \cdot p \cdot (1 - p)}$

Step 1: Calculate the mean.

$\mu = 50 \cdot 0.21 = 10.5$

Step 2: Calculate the standard deviation.

$\sigma = \sqrt{50 \cdot 0.21 \cdot (1 - 0.21)} = \sqrt{50 \cdot 0.21 \cdot 0.79}$

$\sigma = \sqrt{50 \cdot 0.1659} = \sqrt{8.295} \approx 2.881$

Thus:

• Mean = 10.5
• Standard deviation ≈ 2.881

(b) Calculating the probability

We want to calculate the probability that at most 5 individuals in the sample have never been married. Since we are approximating a binomial distribution with a normal distribution, we can use the following formula for the probability:

$P(X \leq 5) \approx P\left( Z \leq \frac{5 - \mu}{\sigma} \right)$

Where:

• $Z$ is the standard normal random variable (Z-score).

Step 1: Calculate the Z-score.

$Z = \frac{5 - 10.5}{2.881} = \frac{-5.5}{2.881} \approx -1.910$

Step 2: Look up the Z-score in the standard normal distribution table or use a calculator to find the probability.

For $Z = -1.910$, the corresponding probability is approximately 0.0281 (this is the area to the left of $Z = -1.910$).

Thus, the probability that at most 5 individuals in the sample have never been married is approximately 0.0281 (rounded to 4 decimal places).

Final Answers:

• (a) Mean = 10.5 individuals
• (a) Standard Deviation ≈ 2.881 individuals
• (b) Probability ≈ 0.0281

Would you like to go over any of the steps in more detail, or do you have any other questions?

Here are some related questions to explore further:

1. How does the Central Limit Theorem apply to binomial distributions?
2. How would the probability change if the sample size was larger or smaller?
3. What is the relationship between the binomial distribution and the normal distribution?
4. How do we calculate probabilities for continuous distributions like the normal distribution?
5. Can we calculate the probability for "at most" using other methods besides the Z-score?

Tip: When approximating a binomial distribution with a normal distribution, remember to use a continuity correction (adjusting the value by 0.5) when you're working with discrete values like 5 or 6, but it's often ignored for large sample sizes.