Final answer:
To find the requested probabilities in a normally distributed variable, calculate the Z-scores and then use a Z-table or calculator to find the area under the normal curve for the calculated Z-scores.
Step-by-step explanation:
The question relates to finding probabilities for a normally distributed variable, specifically the time taken to assemble a component in a factory setting. Given that the time is normally distributed with a mean (μ) of 3.1 minutes and a standard deviation (σ) of 0.6 minutes, we can find the desired probabilities using the standard normal distribution (Z-score).
(a) Probability of taking more than 3.8 minutes: First, calculate the Z-score for 3.8 minutes using the formula Z = (X - μ) / σ, where X is 3.8. Then, use the Z-table or a calculator with normal distribution functions to find the probability that Z is larger than the calculated value.
(b) Probability of taking between 1.8 and 2.5 minutes: Repeat the process for calculating Z-scores, now for X = 1.8 and X = 2.5. The sought probability is the area under the normal curve between these two Z-scores.