Questions: During the busy season it is important for the shipping manager to monitor the model inventory using normal distribution with a mean of 132 minutes and a standard deviation of 15 minutes. Use this table or the AIEX calculator to find the percentage of load times between 110 minutes and 149 minutes according to the model. For your convenience, use the table or the calculator provided. Give your answer to two decimal places. (For example, 8.23%)

Solution Steps

We are given a normal distribution with a mean \( \mu = 132 \) minutes and a standard deviation \( \sigma = 15 \) minutes. We need to find the probability of load times falling between \( 110 \) minutes and \( 149 \) minutes.

To find the probability, we first calculate the Z-scores for the lower and upper bounds of the range:

1. For the lower bound \( 110 \) minutes: \[ Z_{start} = \frac{110 - 132}{15} = -1.4667 \]

2. For the upper bound \( 149 \) minutes: \[ Z_{end} = \frac{149 - 132}{15} = 1.1333 \]

Using the Z-scores, we can find the probability \( P \) that the load times fall between these two values: \[ P = \Phi(Z_{end}) - \Phi(Z_{start}) = \Phi(1.1333) - \Phi(-1.4667) = 0.8002 \]

To express the probability as a percentage, we multiply by \( 100 \): \[ \text{Percentage} = P \times 100 = 0.8002 \times 100 = 80.02\% \]

Final Answer