The problem you've provided is asking to approximate the probability that at most 21 people are left-handed in a sample of 135, given that 16% of the population is left-handed. The problem suggests using a normal approximation to the binomial distribution with a continuity correction.

Let's break it down step by step:

1. Define the binomial distribution parameters:

• Sample size $n = 135$
• Probability of success (left-handedness) $p = 0.16$
• Probability of failure $q = 1 - p = 0.84$
• We are asked to find $P(X \leq 21)$, where $X$ is the number of left-handed people in the sample.

2. Mean and standard deviation of the binomial distribution:

The binomial distribution can be approximated by a normal distribution $N(\mu, \sigma^2)$, where:

• $\mu = np = 135 \times 0.16 = 21.6$
• $\sigma = \sqrt{npq} = \sqrt{135 \times 0.16 \times 0.84} \approx 4.406$

3. Continuity correction:

Since we are approximating a discrete distribution (binomial) with a continuous distribution (normal), we apply a continuity correction. To approximate $P(X \leq 21)$, we instead calculate $P(X \leq 21.5)$ to account for the continuity correction.

4. Convert to standard normal distribution (Z-score):

The Z-score is calculated using the formula: $Z = \frac{X - \mu}{\sigma}$ For $X = 21.5$ (after continuity correction): $Z = \frac{21.5 - 21.6}{4.406} \approx -0.023$

5. Find the probability from the Z-table:

The Z-score of $-0.023$ corresponds to a probability of approximately $0.4908$.

Thus, the probability that at most 21 people in the sample are left-handed is approximately 0.4908, or about 49.08%.

Would you like more details on any step? Here are some related questions:

1. How do you derive the standard deviation in binomial distribution approximations?
2. Why is continuity correction necessary when approximating binomial with normal distribution?
3. Can you explain more about the Z-score and how it's used to find probabilities?
4. What are some conditions under which normal approximation to the binomial is valid?
5. How does the sample size impact the accuracy of the normal approximation?

Tip: Always check that the conditions for using a normal approximation to a binomial distribution are met. Specifically, both $np$ and $nq$ should be greater than 5 for the approximation to be accurate.