8.5 Challenging Integral Applications

Probability Applications

Question #4 (Medium): Normal Distribution

Strategy

Normal distribution has its probability density function as

, where is the mean

and the standard deviation.

Sample Question

Average amount of money spent on groceries for a single adult is shown to be $300 per month with

standard deviation of $80 according to a random city survey, and it is normally distributed.

1) What is the probability that an adult chosen at random spends over $500 per month?

2) What percentage of the city population spend less than $200 on groceries per month?

Solution

Most of survey results can be reflected unto normal distribution. Here, , and , therefore

the pdf is

1) The probability that money spent is over $500 is:

. This cannot be simply evaluated by taking

the integral by hand, but by either plugging into the computer calculator, or looking up normal

distribution table. Therefore,

2) The percentage that the money spent is less than $200 is:

Therefore, the percentage of people spending less than $200 on groceries per month is .

## Document Summary

Normal distribution has its probability density function as. , where is the mean and the standard deviation. Most of survey results can be reflected unto normal distribution. Here, , and , therefore the pdf is: the probability that money spent is over is: the integral by hand, but by either plugging into the computer calculator, or looking up normal distribution table. This cannot be simply evaluated by taking: the percentage that the money spent is less than is: