uppose Abby tracks her travel time to work for 60 days and determines that her mean travel time, in minutes, is 35.6. Assume
§the travel times are normally distributed with a standard deviation of 10.3 min. Determine the travel time x such that 17.36% of the 60 days have a travel time that is at least x.
Give your answer rounded to two decimal places.

To find the travel time \(x\) such that 17.36% of the 60 days have a travel time that is at least \(x\), we need to determine the value of \(x\) where 82.64% (100% - 17.36%) of the travel times are less than \(x\). This problem requires finding a value on the normal distribution corresponding to a certain cumulative probability.

Given:
- Mean (\(\mu\)) = 35.6 minutes
- Standard deviation (\(\sigma\)) = 10.3 minutes
- Cumulative probability = 0.8264 (since 17.36% are greater, 82.64% are less)

We need to find the z-score that corresponds to a cumulative probability of 0.8264. After finding the z-score, we will use it to calculate \(x\).

### Step 1: Find the z-score

Using the cumulative probability, we can find the corresponding z-score from the standard normal distribution table or using a z-score calculator. For a cumulative probability of 0.8264, the z-score is approximately:

\[
z \approx 0.94
\]

### Step 2: Calculate the travel time \(x\)

The formula to convert the z-score to a value \(x\) in the normal distribution is:

\[
x = \mu + z \cdot \sigma
\]

Substituting the known values:

\[
x = 35.6 + 0.94 \times 10.3
\]

Let's calculate the value of \(x\):

\[
x = 35.6 + 9.682 = 45.28 \text{ minutes}
\]

Therefore, the travel time \(x\) such that 17.36% of the 60 days have a travel time that is at least \(x\) is approximately **45.28 minutes**.

Would you like more details, or do you have any questions?

Here are some related questions you might find useful:

1. How do you find the z-score corresponding to a given probability in a normal distribution?
2. What are the implications of having a higher standard deviation in the context of travel times?
3. How would the result change if the mean travel time were different?
4. Can you explain why we subtract the desired percentage from 100% to find the cumulative probability?
5. How does the normal distribution relate to other types of distributions in terms of calculating probabilities?

**Tip:** When dealing with normal distributions, always make sure to check if the distribution is truly normal and understand that extreme outliers can impact the mean and standard deviation significantly.

Suppose Abby tracks her travel time to work for 60 days and determines that her mean travel time, in minutes, is 35.6. Assume the travel times are normally distributed with a standard deviation of 10.3 min. Determine the travel time x such that 17.36% of the 60 days have a travel time that is at least x. Give your answer rounded to two decimal places.

Find the travel time x for Abby's commute where 17.36% of the days have a travel time that is at least x, assuming a normal distribution with a mean of 35.6 minutes and a standard deviation of 10.3 minutes. Understand key concepts like the Z-score and cumulative probability.

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