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Finding the Probability of a Normal Random Variable

Example:  Compute the probability that a randomly selected three-year-old female is between 35 and 40 inches tall.  That is, find

In this exercise, use a normal distribution with  and

Method 1:  To find  press 2^nd [DISTR], select 2:normalcdf( and type in 35, 40 , 35.72 , 3.17 ) and press ENTER.

Method 2:  To find the probability and include a graph, you must first set up the WINDOW so that the graph will be displayed properly.  You will need to set Xmin equal to  and Xmax equal to .  Press WINDOW and set Xmin equal to  by typing in 38.72 – 3 * 3.17.  Press ENTER and set Xmax equal to  by typing in 38.72 + 3 * 3.17.  Set Xscl equal to , which is 3.17.

Setting the Y-range is a little more difficult to do.  A good “rule-of-thumb” is to set Ymax equal to .  For this example, set Ymax = .5/3.17.

Use the blue up arrow to highlight Ymin.  A good value for Ymin is (-) Ymax / 4  so type in

(-) .158 / 4.

Press 2^nd [DRAW] and select 1:ClrDraw and press ENTER ENTER.  Press 2^nd [STATPLOT]and TURN OFF all PLOTS.  Press 2^nd [DISTR], highlight DRAW and select 1:ShadeNorm( and type in  35 , 40 , 38.72 , 3.17 ) and press ENTER.

Conclusion:  The probability that a randomly selected three-year-old female is between 35 and 40 inches tall is .5365 or 53.65%.