In this section, we will discuss the model. For example, let the variable X be the height of an adult American male. Then X is approximately normally distributed. It is centered at 70" (i.e., the mean height is ) and its standard deviation is 4", (i.e. ). For this example, we say that X is normal with mean 70 and variance 16 and we will write it as N(70,16). A picture of the distribution is given in Figure 5.7.
Suppose we want to determine the probability that a man is over 6 feet
tall. That's easy. Just find the area under this curve between 72 and infinity.
What's that, you didn't take calculus? Well no worries. We will often be
finding probabilities like this so we have, of course, a class code
to do so. There are two steps: make a z-score and then choose probability from the analysis menu.
A fact that we need here is that a random variable X is
if and only if the random variable
Note that the distribution of Z does not depend on
To solve our problem, we only need to make a z-score for X
= 72, which is z = (72-70)/4 = .5. Hence, the probability that
X > 72 is the same as the probability that Z > .5. So we only
need to compute the area under the distribution of Z from .5 to
infinity. In terms of the distributions, we want the shaded area in Figure 5.8.
Actually the class code will give us the area to the left of .5. But
recall that the total area under the curve is 1; hence, our answer is
1 - (the area to the left of .5).
To solve this problem with CC, just choose probability from the analysis modules.
You want Cumulative Normal Probabilities and enter .5 at the k-window.
Try it. Remember to get the answer subtract what you see from 1; i.e. the
probability that a man is over 6 foot tall is 1- .6914624612740131 =
Lost? Lets try another one. What's the probability that a man is between 66 and 77 inches tall? Compute the z-scores of 66 and 77. You will get -1 and 1.75. We want the shaded area in Figure 5.9.
Again go to CC and put in 1.75, record the answer. Then put in -1, record the answer. Subtract the second from the first and you have the probability that a man is be between 66 and 77 inches tall. Try it. Hence the answer is .9599 - .1586 = .8013.