---
title: "CLT: average life of 16 bulbs"
subject: "Probability and Statistics"
chapter: 4
key: "semester_5/probability_and_statistics:clt-bulbs-16-850-750-900"
source: https://www.bctengineer.com/q/semester_5/probability_and_statistics/clt-bulbs-16-850-750-900
site: BCT Engineering
---

# CLT: average life of 16 bulbs

Probability and Statistics · Chapter 4: Sampling Distribution

- Asked in: 2073 Chaitra, 6 marks
- Video: [Central Limit Theorem (CLT) | Statistical Inference | VTU Exam Problem | Part 6](https://www.youtube.com/watch?v=aSaOGyPSjbs) — Mathematics Tutor
- Answer checked by Claude Sonnet 5.5 on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_5/probability_and_statistics/clt-bulbs-16-850-750-900
- Whole chapter: https://www.bctengineer.com/md/semester_5/probability_and_statistics/chapter-4

## Question

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.

## 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.

## Answer

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)**

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. Why is x̄ normal here even though n = 16 is small?
   A. The population itself is normal
   B. n = 16 is large enough for the CLT
   C. σ is known
   D. mean equals 800
   Answer: A. If the population is normal, x̄ is exactly normal for every n. (Key point: Why x̄ is normal)

2. The SE of x̄ is:
   A. 40
   B. 10
   C. 2.5
   D. 160
   Answer: B. 40/sqrt(16) = 10 hours. (Key point: Standard error)

3. The z-value for x̄ = 850 is:
   A. 1.25
   B. 5.00
   C. 50
   D. 12.5
   Answer: B. z = (850 − 800)/10 = 5. (Key point: z for 850)

4. P(x̄ < 850) is:
   A. about 0.5
   B. about 0.84
   C. practically 1
   D. about 0.0
   Answer: C. z = 5 puts 850 far into the upper tail, so nearly all the area lies below it. (Key point: Result)

## Close variants set in other exams

- [CLT: bulbs average life below 775 h](https://www.bctengineer.com/question.md?key=semester_5%2Fprobability_and_statistics%3Aclt-bulb-sample-mean-below-775) — Probability and Statistics, chapter 4, asked 2×

## Related in Probability and Statistics

- [CLT: battery cell average life](https://www.bctengineer.com/question.md?key=semester_5%2Fprobability_and_statistics%3Abattery-cells-clt-more-15-less) — Probability and Statistics, chapter 4
- [CLT: average resistance of 25 resistors](https://www.bctengineer.com/question.md?key=semester_5%2Fprobability_and_statistics%3Aclt-resistors-mean-below-95) — Probability and Statistics, chapter 4