The Law of Large Numbers says that if you take samples of larger and larger size
from any population, then the mean
Example 1
A study involving stress is done on a college campus among the students. The stress scores follow a uniform distribution with the lowest stress score equal to 1 and the highest equal to 5. Using a sample of 75 students, find:
- a. The probability that the average stress score for the 75 students is less than 2.
- b. The 90th percentile for the average stress score for the 75 students.
- c. The probability that the total of the 75 stress scores is less than 200.
- d. The 90th percentile for the total stress score for the 75 students.
Let
Problems a and b ask you to find a probability or a percentile for an average or mean.
Problems c and d ask you to find a probability or a percentile for a total or sum.
The sample size,
Since the individual stress scores follow a uniform
distribution,
For problems a and b, let
Problem 1
Find
Solution
The probability that the average stress score is less than 2 is about 0.
normalcdf
Reminder:
Problem 2
Find the 90th percentile for the average of 75 stress scores. Draw a graph.
Solution
Let
Find
The 90th percentile for the average of 75 scores is about 3.2. This means that 90% of all the averages of 75 stress scores are at most 3.2 and 10% are at least 3.2.
invNorm
For problems c and d, let
Problem 3
Find
Solution
The mean of the sum
of 75 stress scores is
The standard
deviation of the
sum of 75 stress
scores is

The probability that the total of 75 scores is less than 200 is about 0.
normalcdf
Reminder:
Problem 4
Find the 90th percentile for the total of 75 stress scores. Draw a graph.
Solution
Let
Find

The 90th percentile for the sum of 75 scores is about 237.8. This means that 90% of all the sums of 75 scores are no more than 237.8 and 10% are no less than 237.8.
invNorm
Example 2
The distribution of ages of statistics students at a certain college has an exponential distribution with a mean age of 22 years. Eighty statistics students are randomly selected. Find
- a. The probability that the average age of the 80 statistics students is more than 20.
- b. The 95th percentile for the average age of the 80 statistics students.
Let
Let
Problem 1
Find
Solution
The probability that the average stress score is more than 20 is 0.7919.
normalcdf
Reminder:
EE key for E.Problem 2
Find the 95th percentile for the average of 75 stress scores. Draw a graph.
Solution
Let
Find
The 95th percentile for the average age of 80 statistics students at a certain community college is about 26.0. This means that 95% of the average ages of statistics students are at most 26.0 and 10% are at least 26.0.
invNorm




