he flight times between two cities can be modeled by a normal distribution with a mean of 75 minutes and a standard deviation of x minutes. Given that 2% of the flight times are longer than 82 minutes. Find the standard deviation

To solve this problem, we need to find the standard deviation, \( x \), given the following conditions:

- The flight times follow a normal distribution with a mean \( \mu = 75 \) minutes.
- 2% of the flight times are longer than 82 minutes.

This can be expressed as:

\[
P(X > 82) = 0.02
\]

In a normal distribution, we can use the z-score formula to standardize this problem:

\[
z = \frac{X - \mu}{\sigma}
\]

Here, \( X = 82 \), \( \mu = 75 \), and \( \sigma = x \). Substituting these values in, we have:

\[
z = \frac{82 - 75}{x}
\]

\[
z = \frac{7}{x}
\]

We know that \( P(X > 82) = 0.02 \) corresponds to a z-score such that 2% of the data lies to the right of it. Using a standard normal distribution table or a calculator, we find that the z-score corresponding to the 98th percentile (since 1 - 0.02 = 0.98) is approximately:

\[
z \approx 2.054
\]

Setting up the equation:

\[
\frac{7}{x} = 2.054
\]

Solving for \( x \):

\[
x = \frac{7}{2.054} \approx 3.41
\]

Therefore, the standard deviation is approximately \( \sigma = 3.41 \) minutes.

Would you like more details on any part of this solution?

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**Related Questions:**

1. How would the problem change if the mean flight time were 80 minutes?
2. What would be the standard deviation if only 1% of flight times were longer than 82 minutes?
3. How can we use the empirical rule to approximate the probability of flight times?
4. What is the z-score for a flight time of 70 minutes with this standard deviation?
5. How would the z-score change if the mean flight time were 70 minutes?

**Tip:** For normal distributions, knowing the mean and standard deviation allows you to calculate probabilities for any range of values.

The flight times between two cities can be modeled by a normal distribution with a mean of 75 minutes and a standard deviation of x minutes. Given that 2% of the flight times are longer than 82 minutes, find the standard deviation.

Learn how to calculate the standard deviation for flight times modeled by a normal distribution with a mean of 75 minutes. This problem explores z-scores and percentiles, using the fact that 2% of flights are longer than 82 minutes. Suitable for students in Grades 10-12.

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