Applications of The Normal Distribution

Suppose that X is a normally distributed random variable with mean mu = 7.0 and standard deviation sigma = 1.1. The probability that X lies between 7.75 and 8.25 is calculated by measuring the area under this normal curve or density function between these two values. As mentioned earlier, finding an area under a curve involves the calculus. How are we non-math types going to measure this area? There are three ways that we can accomplish this task.

Method I. Apply the Standard Normal Distribution

To use this method we convert the X values to z-Scores
The mean mu = 7.0 converts to z = (7.0 - 7.0)/1.1 = 0
The X value 7.75 converts to z = (7.75 - 7.0)/1.1 = 0.69
The X value 8.25 converts to z = (8.25 - 7.0)/1.1 = 1.14.

Looking at a table for the standard normal we calculate the desired area as follows
Area from negative infinity to z = 0.69.6 = 0.7549
Area from negative infinity to z = 1.14 = 0.8729
The area between z =0.69 and z = 1.14 is found by subtracting the two areas.
Therefore the desired area or probability is 0.8729 - 0.7549 = 0.118.

Thus with a normal y distributed random variable of mean 7.0 and standard deviation of 1.1, the probability that a value selected at random lies between 7.75 and 8.25 is 0.118 or about a 12% chance.

Method II. Using the TI-83 Calculator

Area only: Press 2nd > DISTR
Enter 2 (for normalcdf)
Enter 7.75 ( left end point of area)
press comma ,
Enter 8.25 (right endpoint of area)
press comma,
Enter 7.0 (mean of distribution)
press comma,
Enter 1.1 (standard deviation)
close parenthesis,
Press Enter
Answer = 0.119

Method III. Using JAVA APPLETS Found On The Internet

Click here by Sanders and Blond
Click hereby McClelland

Note: If the left end point is negative infinity, use a negative number that is about 10 standard deviations below the mean. If the right end point is positive infinity, use a positive number that is about 10 standard deviations above the mean.