Question

Data for the amount of time that Americans spend commuting to work each day creates a histogram that has a normal shape. The mean is 35 minutes and the standard deviation is 10 minutes. According to the Empirical Rule, which of the following is not a conclusion that we can make?

a. It would be common to find a commuter who takes 43 minutes to commute to work.

b. About 5% of commuters take longer than 55 minutes to commute to work.

c. A commuter who takes 30 minutes to commute to work would have a negative Z-score.

d. A commuter who takes 25 minutes to commute to work is more standard deviations from the mean than a commuter who takes 40 minutes.

Answer 1

By the empirical rule, 68% falls within the first standard deviation (µ ± σ), 95% within the first two standard deviations (µ ± 2σ), and 99.7% within the first three standard deviations (µ ± 3σ).

And this is determined using the Z score: A Z score of 1.00, 2.00 and 3.00 implies that the observation lies 1 (µ ± σ), 2 (µ ± 2σ) and 3 (µ ± 3σ) SDs away from the mean.

A Z score can be computed using the formula:

a. For a commuter who takes 43 minutes to commute to work,

This would come (µ ± σ), where 68% of the observations lie. We may say that it would be common to find a commuter who takes 43 minutes to commute to work. This would lie somewhere less than (µ + σ):

b. P(X> 55)

This is nothing but the probability that the observation lies above 2 SDs from the mean. An observation that lie beyond 2 SDs (i.e. above and below) lies in 100% - 95% = 5% of the data values.

But for Z > 2:

The percentage would be 2.5 + 0.15 = 2.65% 5%

Hence, we cannot say that about 5% of commuters take longer than 55 minutes to commute to work.

c. For X = 30,

Hence, a commuter who takes 30 minutes to commute to work would have a negative Z-score.

d. For X = 25,

A commuter who takes 25 minutes to commute to work is 1 standard deviation away from the mean.

For X = 40,

A commuter who takes 40 minutes to commute to work is 0.5 standard deviation away from the mean.