Question

When Eva commutes to work, the amount of time it takes her to arrive is normally distributed with a mean of 57 minutes and a standard deviation of 4 minutes. Using the empirical rule, determine the interval that represents the middle 95% of her commute times.

Answer 1

The empirical rule, also known as the 68-95-99.7 rule, states that for a normal distribution:

- Approximately 68% of the data falls within one standard deviation of the mean.
- Approximately 95% of the data falls within two standard deviations of the mean.
- Approximately 99.7% falls within three standard deviations of the mean.

Given that Eva's commute times are normally distributed with a mean (\(\mu\)) of 57 minutes and a standard deviation (\(\sigma\)) of 4 minutes, we can determine the interval that represents the middle 95% of her commute times by calculating:

1. The lower bound: \(\mu - 2\sigma\)
2. The upper bound: \(\mu + 2\sigma\)

Calculating the lower bound:

\[
\mu - 2\sigma = 57 - 2(4) = 57 - 8 = 49 \text{ minutes}
\]

Calculating the upper bound:

\[
\mu + 2\sigma = 57 + 2(4) = 57 + 8 = 65 \text{ minutes}
\]

Thus, the interval that represents the middle 95% of Eva's commute times is from \(49\) minutes to \(65\) minutes.

Therefore, the final answer is:

**49 minutes to 65 minutes.**