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CLT: average life of 16 bulbs

Short answer

With μ=800, σ=40 and n=16, the sample mean has SE=10 hours. For less than 850, z=5, probability practically 1; for between 750 and 900, z from −5 to 10, also practically 1.

An electrical firm manufactures light bulbs that have a length of life that is approximately normally distributed with mean equal to 800 hours and standard deviation of 40 hours. Find the probability that a random sample of 16 bulbs will have an average life of (a) less than 850 (b) between 750 to 900.

Life X ~ N(μ = 800, σ = 40) hours, n = 16. The population is normal, so x̄ is normal for any n.

Mean of x̄ = 800; standard error = σ/√n = 40/√16 = 40/4 = 10 hours.

a) Average life less than 850 hours

z = (850 − 800)/10 = 5.00

P(x̄ < 850) = P(Z < 5.00) = 0.5 + 0.5000 (approximately) = 1.0000 (practically 1)

b) Average life between 750 and 900 hours

  • z₁ = (750 − 800)/10 = -5.00
  • z₂ = (900 − 800)/10 = 10.00

P(750 < x̄ < 900) = P(−5 < Z < 10) ≈ 0.5 + 0.5 = 1.000000 (practically 1)

Answer: a) ≈ 1 (0.9999997) b) ≈ 1 (0.9999997)

✓ Answer checked by Claude Sonnet 5.5

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