Let's walk through the problem step-by-step.

Problem Analysis:

We are dealing with a binomial distribution, where:

• $n$ is the number of trials (the total number of eyeglasses examined),
• $p$ is the probability of success (in this case, the probability that an eyeglass is defective),
• The mean $\mu$ (expected value) for a binomial distribution is given by: $\mu = n \cdot p$

Solution:

(a) Find the parameters $n$ and $p$:

• $n = 18$ (since 18 eyeglasses were examined)
• $p = 0.156$ (15.6% defective rate expressed as a decimal)

So: $n = 18, \quad p = 0.156$

(b) Find the mean number of defective eyeglasses: Using the formula for the mean: $\mu = n \cdot p = 18 \cdot 0.156$

$\mu = 2.808$

So, the mean number of defective eyeglasses is: $\boxed{2.81} \quad (\text{rounded to 2 decimals})$

Summary:

• $n = 18$
• $p = 0.156$
• Mean $= 2.81$

Would you like me to explain further or answer anything else?

Here are 5 related questions to explore next:

1. What is the variance formula for a binomial distribution?
2. How do you calculate the probability of exactly 2 defective eyeglasses?
3. What is the standard deviation of the defective eyeglasses in this case?
4. How does increasing $n$ impact the mean and variance?
5. How would you determine the probability of at least 3 defective eyeglasses?

Tip: Always convert percentages to decimals before using them in statistical formulas to avoid calculation errors.