When Marques commutes to work, the amount of time it takes him to arrive is normally distributed with a mean of 37 minutes and a standard deviation of 4.5 minutes. Using the empirical rule, what percentage of his commutes will be between 23.5 and 50.5 minutes?

Answer 1

To find the percentage of his commutes that will be between 23.5 and 50.5 minutes, we first need to calculate the z-scores for each of these times using the formula:

z = (X - μ) / σ

Where:
X = the specific time (23.5 minutes or 50.5 minutes)
μ = the mean (37 minutes)
σ = the standard deviation (4.5 minutes)

For X = 23.5 minutes:
z = (23.5 - 37) / 4.5
z = -13.5 / 4.5
z = -3

For X = 50.5 minutes:
z = (50.5 - 37) / 4.5
z = 13.5 / 4.5
z = 3

Now, we can look up the z-scores in a standard normal table or use a calculator to find the percentage of data that falls between -3 and 3 standard deviations from the mean. The empirical rule states that approximately 99.7% of the data falls within 3 standard deviations of the mean in a normal distribution.

Since -3 to 3 standard deviations cover almost all of the data, we can estimate that approximately 99.7% of Marques' commutes will be between 23.5 and 50.5 minutes.