Question

Dave drives to work each morning at about the same time. His commute time is normally distributed with a mean of 35 minutes and a standard deviation of 5 minutes. The percentage of time that his commute time is less than 44 minutes is equal to the area under the standard normal curve that lies to the ___ of __.

Answer 1

The percentage of time that his commute time is less than 44 minutes is equal to the area under the standard normal curve that lies to the left of 1.8.

Explanation:

Problems of normally distributed samples can be solved using the z-score formula.

In a set with mean
`\mu` and standard deviation
`\sigma`, the zscore of a measure X is given by:

`Z = (X - \mu)/(\sigma)`

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X, or the area of the normal curve to the left of Z. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X, or the area of the normal curve to the right of Z.

In this problem, we have that:

`\mu = 35, \sigma = 5`

Less than 44 minutes.

Area to the left of Z when X = 44. So

`Z = (X - \mu)/(\sigma)`

`Z = (44 - 35)/(5)`

`Z = 1.8`

So the answer is:

The percentage of time that his commute time is less than 44 minutes is equal to the area under the standard normal curve that lies to the left of 1.8.