---
title: "Normal: machinery assembly time"
subject: "Probability and Statistics"
chapter: 3
key: "semester_5/probability_and_statistics:assembly-time-normal-12-9"
source: https://www.bctengineer.com/q/semester_5/probability_and_statistics/assembly-time-normal-12-9
site: BCT Engineering
---

# Normal: machinery assembly time

Probability and Statistics · Chapter 3: Continuous Probability Distributions

- Asked in: 2070 Chaitra, 4 marks
- Answer checked by Claude Sonnet 5.5 on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_5/probability_and_statistics/assembly-time-normal-12-9
- Whole chapter: https://www.bctengineer.com/md/semester_5/probability_and_statistics/chapter-3

## Question

The time required to assemble a piece of machinery is a random variable having approximately a normal distribution with mean 12.9 minutes and standard deviation of 2 minutes. What are the probabilities that the assembly of a piece of machinery of this kind will take (a) at least 11.5 minutes (b) between 11.0 to 14.8 minutes?

## Short answer

a) z=(11.5−12.9)/2=−0.70, P(Z≥−0.70)=0.7580. b) z-scores −0.95 and 0.95, probability = 2×0.3289=0.6578.

## Answer

Assembly time X ~ N(μ = 12.9, σ = 2) minutes.

#### a) At least 11.5 minutes
z = (11.5 − 12.9)/2 = -0.70

P(X ≥ 11.5) = P(Z ≥ −0.70) = 0.5 + P(0 < Z < 0.70) = 0.5 + 0.2580 = **0.7580**

#### b) Between 11.0 and 14.8 minutes
- z₁ = (11.0 − 12.9)/2 = -0.95
- z₂ = (14.8 − 12.9)/2 = 0.95

P(11.0 < X < 14.8) = P(−0.95 < Z < 0.95) = 2 × 0.3289 = **0.6578**

**Answer: a) 0.7580  b) 0.6578**

## Check yourself

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

1. For N(12.9, 2), z at 11.5 minutes is:
   A. +0.70
   B. −0.70
   C. −1.40
   D. −0.35
   Answer: B. z = (11.5 − 12.9)/2 = −0.70. (Key point: z for 11.5)

2. P(X ≥ 11.5) = P(Z ≥ −0.70) equals:
   A. 0.5 − 0.2580 = 0.2420
   B. 0.5 + 0.2580 = 0.7580
   C. 0.2580
   D. 1 − 0.7580
   Answer: B. The area to the right of a negative z includes the whole right half plus the table area. (Key point: At least probability)

3. For 11.0 < X < 14.8, the z-values are −0.95 and 0.95, so the probability is:
   A. 0.3289
   B. 2 × 0.3289 = 0.6579
   C. 0.5 + 0.3289
   D. 1 − 0.3289
   Answer: B. The interval is symmetric about zero, so twice the table area 0.3289. (Key point: Symmetric interval)

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