AP Statistics
Learning the Empirical Rule
(or How my Graphing Calculator Saved Me by Integrating Functions)
Graphing calculator note: since the normal probability density function is a probability function just like
binomial and geometric, the normalpdf is located in the same place [2nd] [VARS].
Input the following command to graph a normal density curve:  = 20  = 3.
Y1 = normalpdf(X,20,3)
Use the command [2nd] [TRACE] [7:  f(x) dx ]
Enter each limit in directly and press [ENTER] after each entry.
This will find the area under the curve. (Round to 3 decimals)
Lower limit: 17
Upper limit: 23
Area:__________
Lower limit: 14
Upper limit: 26
Area:__________
Lower limit: 11
Upper limit: 29
Area:__________
Input the following command to graph a normal density curve:  = 50  = 5.
Y1 = normalpdf(X,50,5)
Lower limit: 45
Upper limit: 55
Area:__________
Lower limit: 40
Upper limit: 60
Area:__________
Lower limit: 35
Upper limit: 65
Area:__________
Input the following command to graph the standard normal density curve:  = 0  = 1.
Y1 = normalpdf(X)
Lower limit: –1
Upper limit: 1
Area:__________
Lower limit: –2
Upper limit: 2
Area:__________
Lower limit: –3
Upper limit: 3
Area:__________
The Empirical Rule:
Approximately_________% of observations is within _____ standard deviation of the mean.
Approximately_________% of observations is within _____ standard deviation of the mean.
Approximately_________% of observations is within _____ standard deviation of the mean.
Empirical Rule Practice
1. At Burnt Mesa Pueblo, archaeological studies have used the method of tree ring dating in an effort to
determine when prehistoric people lived in the pueblo. Wood from several excavations gave a mean
of (year) 1243 with standard deviation 36 years. The distribution of dates is approximately normal.
a. Estimate the range of years centered about the mean in which about 68% of the data will be
found.
b. Estimate a range of years centered about the mean in which about 95% of the data will be
found.
c. Estimate a range of years centered about the mean in which almost all (99.7%of the data will
be found.
2. Assume that the heights of college women are normally distributed, with mean 65 in. and standard
deviation 2.5 in.
a.
b.
c.
d.
e.
f.
g.
What percentage of women are taller than 65 in.?
What percentage of women are shorter than 65 in.?
What percentage of women are between 62.5 in. and 67.5 in.?
What percentage of women are between 60 in. and 70 in.?
What percentage of women are between 60 and 67.5 in?
What percentage of women are shorter than 70 in.?
Approximately how tall would a woman need to be in order to be located in the 50th
percentile? 97th percentile?
3. The incubation time for Rhode Island Red chicks is normally distributed with mean 21 days and
standard deviation approximately 1 day. If 1000 eggs are being incubated, how many chicks do we
expect will hatch
a. In 20 to 22 days?
e. In 20 to 21 days?
b. In 19 to 23 days?
f. In 19 to 20 days?
c. In 18 to 24 days?
g. In 18 to 19 days?
d. In more than 21 days?
h. In 18 days or less?
4. A vending machine automatically pours hot drinks into cups. The amount of hot drink dispensed into
a cup is normally distributed with mean 7.6 oz and standard deviation of 0.4 oz.
a. Estimate the probability that the machine will over flow an 8-oz cup.
b. Estimate the probability that the machine will not over flow an 8-oz cup.
c. The machine has just been loaded with 850 cups. How many of these do you expect will
overflow when served?
d. Estimate the probability that the machine will not over flow an 8-oz cup until the 4th cup.
e. Estimate the probability that if 10 cups are randomly selected, exactly 8 will not over flow.