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To solve this problem, we need to find the probability that a loading time falls between 150 minutes and 174 minutes, given a normal distribution with a mean (\(\mu\)) of 156 minutes and a standard deviation (\(\sigma\)) of 15 minutes. Here are the steps:

1. **Convert Raw Scores to Z-Scores:**
   Use the formula for the Z-score:
   \[
   Z = \frac{X - \mu}{\sigma}
   \]
   where \( X \) is the raw score, \( \mu \) is the mean, and \( \sigma \) is the standard deviation.

   For \( X = 150 \):
   \[
   Z_{150} = \frac{150 - 156}{15} = \frac{-6}{15} = -0.4
   \]

   For \( X = 174 \):
   \[
   Z_{174} = \frac{174 - 156}{15} = \frac{18}{15} = 1.2
   \]

2. **Find the Probabilities Corresponding to Each Z-Score:**
   Using a standard normal distribution table:
   - For \( Z = -0.4 \), the cumulative probability is approximately 0.3446.
   - For \( Z = 1.2 \), the cumulative probability is approximately 0.8849.

3. **Calculate the Probability Between the Two Z-Scores:**
   \[
   P(150 \leq X \leq 174) = P(Z \leq 1.2) - P(Z \leq -0.4) = 0.8849 - 0.3446 = 0.5403
   \]

4. **Convert to Percentage:**
   \[
   0.5403 \times 100 = 54.03\%
   \]

### Final Answer:
The percentage of load times between 150 minutes and 174 minutes is **54.03%**.

Would you like more details on this solution or have any further questions?

Here are some related questions:
1. How would you calculate the probability if the range were between 140 and 160 minutes?
2. What if the mean loading time were different—how would that affect the probability?
3. How can you determine if a normal distribution is a good fit for other datasets?
4. How do you interpret Z-scores in practical scenarios like this one?
5. What are the effects of increasing the standard deviation on the probability range?

**Tip:** Remember that the area under a normal distribution curve between two Z-scores represents the probability within that range.

During the busy season, it is important for the shipping manager at ShipMundo to be able to estimate the time it takes the loading crew to load a truck. The shipping manager has found that she can model the load times using a normal distribution with a mean of 156 minutes and a standard deviation of 15 minutes. Use a table or calculator to find the percentage of load times between 150 minutes and 174 minutes according to the model.

Determine the percentage of load times between 150 and 174 minutes based on a normal distribution with a mean of 156 minutes and a standard deviation of 15 minutes. This problem explains how to calculate Z-scores and interpret the results using a standard normal distribution table.

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