Question

eyeglassomatic manufactures eyeglasses for different retailers. in march 2010, they tested to see how many defective lenses they made, and there were 15.2% defective lenses due to scratches. suppose eyeglassomatic examined 12 pairs of eyeglasses.
a) state the random variable.
select an answer
b) list the given numbers with correct symbols.
? = 12
? = 0.152
c) explain why this is a binomial experiment. check all that apply.

Explanation:

Step1: Identify the random variable

The random variable is the number of defective lenses in the 12 pairs of eyeglasses examined.

Step2: List binomial symbols

In a binomial experiment, we have \(n\) (number of trials), \(p\) (probability of success), \(q\) (probability of failure), and \(X\) (random variable). Here, \(n = 12\) (12 pairs of eyeglasses), \(p=0.152\) (probability of a lens being defective), \(q = 1 - p=1 - 0.152 = 0.848\), and \(X\) is the number of defective lenses.

Step3: Check binomial conditions

1. Fixed number of trials: There are \(n = 12\) pairs of eyeglasses (fixed number of trials).
2. Independent trials: The defectiveness of one pair of eyeglasses (lenses) is assumed to be independent of another pair.
3. Two - outcome per trial: Each pair of eyeglasses (lenses) is either defective (success) or non - defective (failure).
4. Constant probability of success: The probability of a pair of eyeglasses (lenses) being defective \(p = 0.152\) is constant for all 12 pairs.

Answer:

a) The random variable \(X\) is the number of defective lenses in the 12 pairs of eyeglasses.
b) \(n = 12\), \(p=0.152\), \(q = 0.848\), \(X\) (number of defective lenses).
c) It is a binomial experiment because:

• There is a fixed number of trials (\(n = 12\)).
• Trials are independent.
• Each trial has two outcomes (defective or non - defective).
• The probability of success (\(p = 0.152\)) is constant.