Math Problem Statement

An eyeglass manufacturer knows from past experience that 15.6% of manufactured lenses are defective due to scratches. Suppose that the manufacturer examined eighteen eyeglasses. Assume that the binomial distribution applies. (a) Find the parameters n and p. (b) Find the mean number of eyeglasses that are defective due to scratches. Round to 2 decimals.

Solution

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

Problem Analysis:

We are dealing with a binomial distribution, where:

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

Solution:

(a) Find the parameters nn and pp:

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

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


(b) Find the mean number of defective eyeglasses: Using the formula for the mean: μ=np=180.156\mu = n \cdot p = 18 \cdot 0.156

μ=2.808\mu = 2.808

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


Summary:

  • n=18n = 18
  • p=0.156p = 0.156
  • Mean =2.81= 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 nn 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.

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Binomial Distribution
Probability
Mean (Expected Value)

Formulas

Mean (μ) = n × p

Theorems

Binomial Distribution Theorem

Suitable Grade Level

Grades 10-12