The lives of 2000 electric bulbs are normally distributed with mean 1000 hours and standard deviation 100 hours. The number of bulbs expected to have a life outside the range 800 hours to 1200 hours is approximately (Given \(P(0 \le Z \le 2) = 0.4772\), where Z is the standard normal variate)
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → 91
The standard normal variates of the two limits are
\( \displaystyle Z = \frac{800 - 1000}{100} = -2,\quad Z = \frac{1200 - 1000}{100} = 2 \)
Each tail beyond 2 (or below −2) has area 0.5 − 0.4772 = 0.0228. Life outside the range means either tail:
\(P = 2 \times 0.0228 = 0.0456\)
Expected number out of 2000:
\(2000 \times 0.0456 = 91.2 \approx 91\)
Why the other options are incorrect
Option (1) 46 — Incorrect. This is \(2000 \times 0.0228\), only one of the two tails.
Option (2) 954 — Incorrect. This is \(2000 \times 0.4772\), the bulbs lasting between 1000 and 1200 hours.
Option (4) 1909 — Incorrect. This is \(2000 \times 0.9544\), the bulbs inside the range.