Download Empirical Rules An empirical rule in statistics is a

Empirical Rules
An empirical rule in statistics is a rule of thumb. Two of the most commonly used
empirical rules in statistics concern the intervals:
x ± σ and x ± 2σ
If the data can be represented by a standard normal distribution then the first interval
contains about 68% of the values, and the second interval contains about 95 % of the
values. As the distribution deviates from a standard normal distribution, these empirical
rules become less applicable. These rules can help make a quick determination about the
exceptionality of a given value.
For example, let’s say that test scores on the final exam are approximately normally
distributed with a mean of 71 and standard deviation of 5. This means that the mean and
the median are the same number and the probability histogram would be a symmetric
distribution looking something like this:
Continuing with this example, let’s say a friend of yours named Vinnie scored a 74.
What you could say about his score is that it is less than one standard deviation from the
mean. Using one of the empirical rules, you could place Vinnie with the middle 68% of
the of the students whose scores where less than one standard deviation from the mean.
His score is not exceptional.
Another friend named Suzy scores an 84. Her score is more than two standard deviations
from the mean. The empirical rule says that 95% of the scores are between two standard
deviations from the mean. The implication is that she is part of the special group of 5%
of the students who are more than two standard deviations from the mean. More
specifically, since she scored on the high end, she is part of a special group that contains
2.5% of the highest scores. By this definition, she is exceptional.
Problems.
1. As a highway engineer, you are studying traffic patterns on a certain stretch of Highway 12.
During the period from 6:00 AM to 8:00 AM the speeds of vehicles using this patch are
approximately normally distributed with a mean of 75 and standard deviation of 4.
a. Give the interval that would contain 68% of the speeds of vehicles on that stretch of
highway.
b. Give the interval that would contain 95% of the speeds of vehicles on that stretch of
highway.
c. The speed limit on this stretch of highway is 65 mph. Discuss the possibility of
finding drivers driving at or below the speed limit during this time of the day.
2. A coin is flipped 1000 times.
a. What would be the expected number (average) of heads out of 1000 flips?
b. If these 1000 flips were repeated numerous times, what would be the standard
deviation of the number of heads?
c. Make an interval within which you would expect to find 68% of the number of heads
out of 1000 flips.
d. Make an interval within which you would expect to find 95% of the number of heads
out of 1000 flips.
3. Another empirical rule deals with three standard deviations x ± 3σ
a. What percent of scores of scores will be within this interval?
b. Why would an empirical rule using 4 standard deviations not be very useful?