Section 5.2 Normal Distributions: Finding Probabilities

Example 1 A survey indicates that people use their computers an average of 2.4 years before upgrading to a new machine. The standard deviation is 0.5 year. A computer owner is selected a random. Find the probability that he or she will use it for less than 2 years before upgrading. Assume that the variable x is normally distributed.  There is a 21.19% chance owners will upgrade in less than 2 years.

Example 2 A Ford Focus manual transmission gets an average of 27 miles per gallon (mpg) in city driving with a standard deviation of 1.6 mpg. A Focus is selected at random. What is the probability that it will get more than 31 mpg? Assume that gas mileage is normally distributed.  There is a.62% chance the car gets more than 31 mpg.

Example 3  A survey indicates that for each trip to the supermarket, a shopper spends an average of µ = 45 minutes with a standard deviation of σ = 12 minutes. The length of time spent in the store is normally distributed and is represented by the variable x. A shopper enters the store. (a.) Find the probability that the shopper will be in the store for each interval of time listed below. (b.) If 200 shoppers enter the store, how many shoppers would you expect to be in the store for each interval of time listed below?  Between 24 and 54 minutes  There is a 73.33% chance that a person will be in the store between 24 and 54 minutes.  More than 39 minutes  There is a 69.15% chance that a person will be in the store for more than 39 minutes.  Between 33 and 60 minutes  There is a 73.57% chance that a person will be in the store between 33 and 60 minutes.

Example 4 Assume that cholesterol levels of men in the US are normally distributed, with a mean of 215 milligrams per deciliter and a standard deviation of 25 milligrams per deciliter. You randomly select a man from the US. What is the probability that his cholesterol level is less than 175? Use a TI-83 to find the probability.  There is a 5.48% that a man will have a cholesterol level less than 175 mpd.  A man from the US is selected at random. What is the probability that his cholesterol is between 190 and 225?  There is a 49.68% chance that a man will have a cholesterol level between 190 and 225 mpd.

TOTD  The time per week a student uses a lab computer is normally distributed, with a mean of 6.2 hours and a standard deviation of 0.9 hour. A student is randomly selected.  Find the probability that the student uses a lab computer less than 4 hours per week.  Find the probability that the student uses a lab computer between 4 and 7 hours per week.  Find the probability that the student uses a lab computer more than 7 hours per week.