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The proportion of travel times that are at least  𝑥  is represented symbolically as  𝑃(𝑋≥𝑥) .
𝑃(𝑋≥𝑥)=0.2611
In order to find  𝑥 , first find the value 𝑧  such that 26.11% of the total area under the standard normal density curve is to the right of  𝑧 . You can find  𝑧  using this standard normal distribution table or this list of software manuals.

The  𝑧 ‑score formula, where  𝜇  is the mean and  𝜎  is the standard deviation, can be used to convert  𝑧  into travel times.
𝑧=𝑥−𝜇𝜎

Be sure to round your answer correctly to two decimal places.

To solve this problem, we need to find the travel time \( x \) such that the proportion of travel times that are at least \( x \) is 0.2611. In other words, we need to determine \( x \) such that \( P(X \geq x) = 0.2611 \). Here's the step-by-step process:

### Step 1: Find the \( z \)-score
We know that the proportion to the right of \( z \) is 0.2611. In a standard normal distribution, this means the area to the left of \( z \) (i.e., \( P(Z \leq z) \)) is \( 1 - 0.2611 = 0.7389 \).

We now look up the value of \( z \) in a standard normal distribution table corresponding to a cumulative probability of 0.7389. From the table (or using statistical software), the closest \( z \)-score is approximately **0.62**.

### Step 2: Use the \( z \)-score formula
We now convert the \( z \)-score back to the travel time \( x \) using the formula:
\[
z = \frac{x - \mu}{\sigma}
\]
Where:
- \( z \) is the standard normal score (which we found to be 0.62),
- \( \mu \) is the mean travel time,
- \( \sigma \) is the standard deviation of the travel times.

Rearranging the formula to solve for \( x \):
\[
x = \mu + z \cdot \sigma
\]
Substitute \( z = 0.62 \), and you'll need the values of \( \mu \) and \( \sigma \) to complete the calculation. Once you have these, plug them in and calculate \( x \).

### Step 3: Round the result
After calculating \( x \), round the final answer to two decimal places as per the instructions.

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Do you have the values for \( \mu \) and \( \sigma \) to finish the calculation?

### 5 related questions:
1. How is the standard normal distribution used in real-world problems?
2. What are some common methods for finding the area under the normal distribution curve?
3. How does the \( z \)-score formula relate to data standardization?
4. Can you explain why the area to the right of \( z \) represents probabilities?
5. How do different types of tables (cumulative or non-cumulative) affect the interpretation of the \( z \)-score?

**Tip:** Always check whether a table gives cumulative probabilities from the left or right, as this affects your \( z \)-score lookup.

Sorry, that's incorrect. The proportion of travel times that are at least 𝑥 is represented symbolically as 𝑃(𝑋≥𝑥). 𝑃(𝑋≥𝑥)=0.2611. In order to find 𝑥, first find the value 𝑧 such that 26.11% of the total area under the standard normal density curve is to the right of 𝑧. You can find 𝑧 using this standard normal distribution table or this list of software manuals. The 𝑧‑score formula, where 𝜇 is the mean and 𝜎 is the standard deviation, can be used to convert 𝑧 into travel times. 𝑧=𝑥−𝜇𝜎. Be sure to round your answer correctly to two decimal places.

Learn how to solve for the travel time x such that P(X ≥ x) = 0.2611 using the z-score formula and standard normal distribution. This problem demonstrates how to find the corresponding z-score and calculate x using the mean and standard deviation. Suitable for Grades 10-12.

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