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Honors Math 3
Unit 3: Inferential Statistics
Name_______________________________
Date________________________________
Central Limit Theorem (Defined in section 3.16)
Warm-up
Consider spinner with mean m and standard deviation s . If the spinner is spun n times and the results added,
how do the mean and standard deviation change?
Exploration
1. A spinner has the number 1, 3, 5 and 7 on it. What is the mean and standard deviation for one spin? Leave
your standard deviation as a radical.
2. The spinner will be spun twice and the results are averaged. The following table gives the sample space for
the possible averages. What is the mean and standard deviation for this sample space? Again, leave your
standard deviation as a radical.
average
1
3
5
7
1
3
5
7
1
2
3
4
2
3
4
5
3
4
5
6
4
5
6
7
3. How did the mean and standard deviation change from one spin to averaging two spins?
Generalize:
Consider spinner with mean m and standard deviation s . If the spinner is spun n times and the results
averaged, how do the mean and standard deviation change?
Central Limit Theorem: Let X be a random variable with mean m and standard deviation s . The distribution
(
)
for the sum of the outputs of X over n trials is more and more closely approximated by N m n, s n as n
grows larger.
Second Version of CLT: Let X be a random variable with mean m and standard deviation s . The distribution
(
for the average of the outputs of X over n trials is more and more closely approximated by N m,
s
n
) as n
grows larger.
Examples:
A Nielson report stated the children between the ages of 2 and 5 watch an average of 25 hours of television
per week. Assume this variable is normally distributed with a standard deviation of 3 hours.
1. If 20 children between the ages of 2 and 5 are randomly selected, what is the probability that total number
of hours of television they watch per week is between 490 and 520 hours?
2. If 20 children between the ages of 2 and 5 are randomly selected, what is the probability that they watch
an average of 26.3 hours of television per week?
Homework
1. A special number cube has a mean of 5 and standard deviation of 2 for one roll.
a. The number cube is rolled 20 times and the results added together. What is the probability that the
sum is exactly 90?
b. The number cube is rolled 20 times and the results added together. What is the probability that the
sum is between 80 and 110?
c. The number cube is rolled 20 times and the results averaged. What is the probability that the average
is between 4.5 and 7?
2. For a certain brand of light bulbs, each bulb has a mean life expectancy of 675 hours and a standard
deviation of 50 hours.
a. What is the probability of a case of 12 light bulbs lasting an average of 660 hours?
b. What is the probability of a case of 12 light bulbs lasting an average of between 670 and 700 hours?
c. What is the probability of a case of 12 light bulbs lasting a total of between 8,000 and 8,300 hours?
3. A 50-question multiple-choice test has 5 options for each question.
a. If someone randomly guesses on each question, what is the probability that they get a score between 8
and 11?
b. Twenty people took this test and randomly guessed on each question. What is the probability that
their average score is between 8 and 11?
>>In case you missed this, here’s a reminder about z-scores:
A z-score, or standardized score, converts a value from a normal distribution with mean m and standard
deviation s into a value on the unit normal distribution (with mean 0 and standard deviation 1). In other
words, it indicates how many standard deviations above (or below) the mean a particular value falls. It allows
us to compare various data sets.
x-m
z - score =
s