Question

eyeglassomatic manufactures eyeglasses for different retailers. in march 2010, they tested to see how many defective lenses they made, and there were 15.28 defective lenses due to scratches. suppose eyeglassomatic examined 12 pairs of eyeglasses.
a) state the random variable.
b) list the given numbers with correct symbols.
c) explain why this is a binomial experiment. check all that apply.

Explanation:

Part (a)

Step1: Define Random Variable

The random variable \( X \) represents the number of 12 randomly selected pairs of eyeglasses that are scratched (or have defective lenses due to scratches). So, we state what \( X \) measures in the context.

Step1: Identify \( n \)

The number of trials (pairs of eyeglasses examined) is \( n = 12 \), as we are testing 12 pairs.

Step2: Identify \( p \)

The proportion of defective lenses (due to scratches) is calculated from the data. The percentage of defective lenses is \( 15.28\% \), so \( p=\frac{15.28}{100} = 0.1528\approx0.152 \) (matching the given value). And \( x \) would be the number of successes (scratched pairs), but here we list \( n = 12 \), \( p\approx0.152 \).

Step1: Binomial Experiment Conditions

A binomial experiment has four conditions: (1) Fixed number of trials (\( n \) is fixed, here \( n = 12 \)), (2) Independent trials (assuming each pair of eyeglasses is independent of others), (3) Two outcomes (success: scratched/defective, failure: not scratched/defective), (4) Constant probability of success (\( p \) is constant, here \( p\approx0.152 \) for each trial).

Step2: Verify Each Condition

• Fixed trials: We have \( n = 12 \) pairs, so fixed.
• Independent: Assuming each pair's scratch status is independent (reasonable if manufacturing/scratching is random).
• Two outcomes: A pair is either scratched (defective) or not.
• Constant \( p \): The probability of a pair being scratched is the same for each pair (based on the overall defect rate).

Answer:

The random variable \( X \) is the number of 12 randomly selected pairs of eyeglasses that have defective lenses due to scratches (or are scratched).

Part (b)