To solve these probability questions, we'll use the properties of the normal distribution and the given mean (μ) and standard deviation (σ).
Given data:
Mean (μ) = 6.7 minutes
Standard deviation (σ) = 2.2 minutes
(a) Probability that the assembly time will be 5 minutes or less:
We need to find the area under the normal curve to the left of 5 minutes.
Z-score = (X - μ) / σ
Z-score for X = 5 minutes:
Z = (5 - 6.7) / 2.2 ≈ -0.773
Using a standard normal distribution table or calculator, we find the cumulative probability corresponding to Z = -0.773 is approximately 0.2206.
Therefore, the probability that the assembly time will be 5 minutes or less is approximately 0.2206 or 22.06%.
(b) Probability that the assembly time will be 10 minutes or more:
We need to find the area under the normal curve to the right of 10 minutes.
Z-score for X = 10 minutes:
Z = (10 - 6.7) / 2.2 ≈ 1.5
Using the standard normal distribution table or calculator, we find the cumulative probability corresponding to Z = 1.5 is approximately 0.9332.
To find the probability of 10 minutes or more, we subtract the cumulative probability from 1:
P(X ≥ 10) = 1 - 0.9332 ≈ 0.0668
Therefore, the probability that the assembly time will be 10 minutes or more is approximately 0.0668 or 6.68%.
(c) Probability that the assembly time will be between 5 and 10 minutes:
We need to find the area under the normal curve between 5 and 10 minutes.
Z-score for X = 5 minutes (from part a):
Z = -0.773
Z-score for X = 10 minutes (from part b):
Z = 1.5
Using a standard normal distribution table or calculator, we find the cumulative probabilities corresponding to Z = -0.773 and Z = 1.5:
P(X ≤ 5) ≈ 0.2206
P(X ≥ 10) ≈ 0.0668
To find the probability between 5 and 10 minutes, we subtract the cumulative probabilities:
P(5 < X < 10) = P(X ≥ 10) - P(X ≤ 5) ≈ 0.0668 - 0.2206 ≈ 0.1538
Therefore, the probability that the assembly time will be between 5 and 10 minutes is approximately 0.1538 or 15.38%.