Question

The time required for a citizen to complete the 2010 U.S. Census "long" form is normally distributed with a mean of 40 minutes and a standard deviation of 10 minutes. The lowest 10 percent of the citizens would need at least how many minutes to complete the form

Answer 1

The lowest 10 percent of the citizens would need at least 52.8 minutes to complete the form.

Explanation:

Problems of normally distributed samples are 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. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.

In this problem, we have that:

`\mu = 40, \sigma = 10`

The slowest 10 percent of the citizens would need at least how many minutes to complete the form

This is the value of X when Z has a pvalue of 0.9. So it is X when Z = 1.28.

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

`1.28 = (X - 40)/(10)`

`X - 40 = 1.28*10`

`X = 52.8`

The lowest 10 percent of the citizens would need at least 52.8 minutes to complete the form.