To solve this problem, we need to use the binomial distribution because we're dealing with a fixed number of trials (14 people), and each trial has two possible outcomes: being older than 60 (success) or not (failure).

Let’s break it down step by step:

Step 1: Define the parameters

• The probability of being older than 60, $p = 0.79$.
• The number of trials, $n = 14$.
• We want to find the probability that at least 11 people in the group of 14 are older than 60, so we need the cumulative probability for $X \geq 11$, where $X$ is the number of people older than 60.

Step 2: Binomial probability formula

The binomial probability mass function (PMF) is given by:

$P(X = k) = \binom{n}{k} p^k (1 - p)^{n-k}$

where $\binom{n}{k}$ is the binomial coefficient, representing the number of ways to choose $k$ successes from $n$ trials.

Step 3: Calculate $P(X \geq 11)$

We need to compute the probability of having 11 or more people older than 60. This can be written as:

$P(X \geq 11) = P(X = 11) + P(X = 12) + P(X = 13) + P(X = 14)$

Step 4: Use a binomial calculator or normal approximation

Calculating the individual binomial probabilities for $X = 11, 12, 13, 14$ can be tedious by hand, so it's common to use a calculator or statistical software. Alternatively, we can approximate using a normal distribution since the binomial distribution for large $n$ can be approximated by a normal distribution.

Approximation:

For a binomial distribution with parameters $n = 14$ and $p = 0.79$, the mean $\mu$ and standard deviation $\sigma$ are:

$\mu = np = 14 \times 0.79 = 11.06$ $\sigma = \sqrt{np(1 - p)} = \sqrt{14 \times 0.79 \times 0.21} \approx 1.77$

Using a normal approximation, we want to find $P(X \geq 11)$, which is equivalent to finding the area to the right of $X = 10.5$ (since we use a continuity correction). We can standardize this value to a z-score:

$z = \frac{10.5 - 11.06}{1.77} \approx -0.32$

Using the standard normal distribution table or a calculator, we find:

$P(Z \geq -0.32) \approx 0.625$

Step 5: Conclusion

After evaluating the possible options, the closest match to this calculation is 0.66, so the correct answer is:

b. 0.66

---

Would you like more details on this solution or have any further questions?

Here are some related questions for you:

1. How do you calculate the binomial probability $P(X = k)$ manually?
2. What is the continuity correction, and why do we use it in normal approximation?
3. How would the result change if the probability $p$ were smaller (say 0.5)?
4. What are the assumptions of using a normal approximation to a binomial distribution?
5. How do you interpret a z-score in terms of probability?

Tip: When working with binomial distributions, if the number of trials $n$ is large and $p$ is not too close to 0 or 1, you can often approximate the binomial with a normal distribution for easier calculations.