Question

When Aria commutes to work, the amount of time it takes her to arrive is normally distributed with a mean of 28 minutes and a standard deviation of 4.5 minutes. Out of the 262 days that Aria commutes to work per year, how many times would her commute be between 32 and 35 minutes, to the nearest whole number?

Answer 1

Answer:

Her commute would be between 32 and 35 minutes 33 times.

Step-by-step 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:

$Z = \frac{X - \mu}{\sigma}$

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:

$\mu = 28, \sigma = 4.5$

Proportion of days in which the commute is between 32 and 35 minutes:

This is the pvalue of Z when X = 35 subtracted by the pvalue of Z when X = 32.

X = 35

$Z = \frac{X - \mu}{\sigma}$

$Z = \frac{35 - 28}{4.5}$

$Z = 1.555$

$Z = 1.555$ has a pvalue of 0.94.

X = 32

$Z = \frac{X - \mu}{\sigma}$

$Z = \frac{32 - 28}{4.5}$

$Z = 0.89$

$Z = 0.89$ has a pvalue of 0.8133.

0.94 - 0.8133 = 0.1267

Out of 262 days:

Each day, 0.1267 probability

0.1267*262 = 33

Her commute would be between 32 and 35 minutes 33 times.