Normal Distribution

One of the most important continuous probability models is the normal probability model . It is important because of the Central Limit Theorem , which we will discuss in the next chapter. Briefly, the Central Limit Theorem says that if you add up a bunch of random errors (independent errors under identical conditions) the distribution of this sum is approximately bell shaped .

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 $\mu=70''$) and its standard deviation is 4", (i.e. $\sigma = 4''$). 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.

  

Figure 5.7: A N(70,16) distribution

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 $N(\mu,\sigma^2)$ if and only if the random variable $Z=\frac{X - \mu}{\sigma}$ is N(0,1). Note that the distribution of Z does not depend on $\mu$ and $\sigma$.

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.

  

Figure 5.8: P(Z>.5)

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 = .3086.

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.

  

Figure 5.9: P(-1 < Z < 1.75)

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.

Exercise 6.3.1  

1.

Consider the height example given above. Determine the probability that a man is between 63 and 71 inches tall. (Ans: 0.5586).

2.

Suppose the passing grade on a standardized exam is 450. Suppose we know that scores on this exam are approximately normally distributed with mean 430 and standard deviation 50.

3.

Find the probability that a person passes the exam. (Ans:.3445).

4.

Find the probability that a person scores over 500. (Ans: .0807).

5.

Find the probability that a person scores between 400 and 480. (Ans: .5670) .

6.

Suppose a part is acceptable only if it is less that .1" long. Suppose the lengths of these parts are approximately normally distributed with mean .09" and standard deviation .018". Find the probability that a part is acceptable. (Ans: .7107).

7.

In the last problem, suppose we make 20 of these parts. Find the probability that at least 10 are acceptable. (Ans: .9870).