Download Normal Probability Distributions

Chapter 6 –Normal Probability Distributions
The normal distribution is a special type of bell-shaped curve.
Defn: A random variable X is said to be normally distributed or to have a normal distribution if its
distribution has the shape of a normal curve:
f x  
1
2 

e
 x   2
2 2
, for -  < x < , -  <  < , and  > 0.
Here  and  are the parameters of the distribution;  = the mean of the random variable X (or of the
probability distribution); and  = the standard deviation of X. The two mathematical constants
appearing in the formula,   3.141592654, and e  2.718281828.
The normal distribution is not just a single distribution, but rather a family of distributions; each
member of the family is characterized by a particular pair of values of  and .
The mean, , tells us the location of the center of the bell-shaped curve. The standard deviation, ,
tells us how wide the bell-shaped curve is; smaller values of  denote narrower, more sharply peaked
bell-shaped curves; larger values of  denote wider, less sharply peaked bell-shaped curves. In
particular, the points given by  -  and  +  are the inflection points of the curve.
Characteristics of normal distributions:
1) The bell is symmetric around the mean value .
2) The standard deviation, , describes the width of the bell.
3) The distribution is unimodal (one peak), and the mean, median and mode are all equal.
4) The curve is continuous; i.e., there are no gaps.
5) For any two numbers, a and b, the probability that the random variable X will be found to have a
value between a and b is denoted by P(a  X  b), and is the area under the curve above the interval
from a to b. (Note that the probability is concentrated around ). This probability can be found using
a technique of calculus, called numerical integration.
6) The total area under the curve is 1.
7) The curve extends indefinitely in both directions, approaching, but never touching, the horizontal
axis.
The Empirical Rule:
For the normal distribution,
1) The probability that X will be found to have a value between  -  and  +  is approximately 0.68;
2) The probability that X will be found to have a value between  - 2 and  + 2 is approximately
0.95;
3) The probability that X will be found to have a value between  - 3 and  + 3 is approximately
0.9974.
Some examples of variables which have approximately normal distributions are:
1) Heights of people in the general population of adults;
2) Diameters of pine tree trunks at 3 ft. above the ground;
3) Scores on an IQ test;
4) Fill weights of 12-oz cans of Pepsi-Cola.
There is a particular, important member of the family, called the standard normal distribution, which
has  = 0 and  = 1, so that
f z  
2
1  z2
e .
2
We use a different notation for a standard normal random variable, calling it Z, rather than X.
The following fact is one of the reasons that the standard normal distribution is so important: If X has
a normal distribution with mean  and standard deviation , then the random variable
X 
has a standard normal distribution.
Z 

This means that if we have a random variable X with a Normal(, ) distribution, and we want to find
the probability that X will be found to have a value between the numbers a and b, we can first convert
to a standard normal distribution, and 1) find the desired probability in a table of the standard normal
distribution or 2) find the probability using the TI-83 calculator.
To find areas under a normal distribution curve (probabilities) using the Table 3 in Appendix B (p.
504):
1) Convert the probability statement to a statement about a standard normal distribution.
2) Draw a picture, shading the desired area.
3) Use the properties of the normal distribution to get the desired probability in terms of a probability
that Z will take on a value between 0 and some positive number.
4) Go to Table 3 and look up that positive number, finding the probability from step 3) in the body of
the table.
5) Calculate the desired probability.
To find areas under a normal distribution curve (probabilities) using the TI-83 calculator:
1) Choose 2nd, DISTR, and normalcdf(.
2) Enter the left-hand endpoint of the interval, the right-hand endpoint of the interval, the mean, , of
the normal distribution (can skip this if the distribution is standard normal), and the standard deviation,
, of the interval (can skip this if the distribution is standard normal).
3) Hit ENTER. The desired probability appears.
The following examples involve finding probabilities associated with a random variable Z which has a
standard normal distribution.
Example: p. 244, Exercise 6.11.
Example: p. 244, Exercise 6.15.
Example: p. 244, Exercise 6.20.
The normal distribution occurs in many situations in nature, in manufacturing processes, in many
instances of analysis of survey data, and in the data analysis for most scientific experiments. In the
examples below, we are given situations in which the random variable of interest, X, has a normal
distribution for which we know the mean, , and the standard deviation, . In most real-world
applications, we do not know the values of  and , but must estimate the values using sample data.
We will talk about techniques of estimation when we take up the topic of statistical inference.
Note: When calculating probabilities associated with random variables, please write complete
probability statements; it will help to prevent confusion and mistakes. For example, if we want the
probability that a random variable X will be found to have a value between the numbers a and b, we
would write P(a  X  b).
Example: p. 255, Exercise 6.45
a) P(X > 250) = ?
b) P(X < 150) = ?
Example: p. 255, Exercise 6.47
a) P($11.00 < X < $15.00) = ?
b) P($8.00 < X < $19.00) = ?
c) P(X > $ 20.00) = ?
d) P(X < $6.00) = ?
Example: p. 255, Exercise 6.48
a) P(X > 32.02) = ?
b) What two probability distributions do we need to use here?
When we talk about statistical inference, we will need to invert the above process of finding
probabilities. Instead of finding the probability associated with a given interval of values of a normally
distributed random variable, we will need to find one or more values of the random variable which are
the endpoints of an interval for which we are given the probability. To do this, we use the invNorm(
function of the TI-83 calculator. In the examples below, we are given the mean, , and the standard
deviation, , of the normal distribution of the random variable, X.
Example: p. 262, Exercise 6.63
a)
To find the quantity z using the TI-83 calculator:
1) Enter 2nd, DISTR, invNorm(.
2) We are looking at an interval of the form (z, ). Enter 1 – 0.03 in the calculator, the mean, 0, of the
distribution and the standard deviation, 1, of the distribution, and press ENTER.
3) The result is z = 1.8808.
c)
1) Enter then 2nd, DISTR, invNorm(.
2) We are looking at an interval of the form (z, ). Enter 1 - 0.75 in the calculator, the mean, 0, of the
distribution, and the standard deviation, 1, of the distribution, and press ENTER.
3) The result is –0.6745.
Example: p. 263, Exercise 6.67