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A manufacturer finds 3% of its Christmas light bulbs defective. For a string of 100 lights, the Poisson model with μ=3 gives P(x≤4)=0.8153, while the binomial model X~B(100,0.03) gives P(x≤4)=0.8179. What can be concluded about the size of this difference?
AThe Poisson approximation is very good, off by only 0.0026
BThe Poisson approximation fails badly here
CThe two models disagree by more than 0.50
DThe binomial model cannot be computed for this case
Answer & Solution
Correct answer: A. The Poisson approximation is very good, off by only 0.0026
1. Subtract the two computed probabilities: 0.8179 - 0.8153 = 0.0026.
2. This gap of 0.0026 is extremely small on a probability scale from 0 to 1.
3. A gap this small counts as a very good approximation, matching that description.
4. Options B, C, and D all overstate or misrepresent how close the two models actually land.
_Source: OpenStax Introductory Statistics (CC BY 4.0), Ch 4 "Discrete Random Variables", section 4.6 Poisson Distribution_
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