Download Cents and the Central Limit Theorem

Cents and the Central Limit Theorem
Name:________________
Date:__________ Per:___
SCENARIO: Many of the variables you have studied so far have had normal distributions. Many
distributions, however, are not normal or any other standard shape.
QUESTION: If the shape of a distribution is not normal, can we make any inferences about the
mean of a random sample from that distribution?
OBJECTIVE: In this activity you will discover the central limit theorem (CLT) by observing the
shape, mean, and standard deviation of the sampling distribution of the mean for samples taken
from a distribution that is decidedly not normal.
PRIOR TO ACTIVITY: We have collected many pennies for this activity and recorded their
ages where a penny from 2010 has an age of zero, a penny from 2009 has an age of one, and so
on. The total number of pennies from each year has been recorded and the true mean of this
population has been found.
ACTIVITY
1. What do you think the shape of the distribution of all the ages of the pennies from the
students in your class will look like? Why?
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2. Use the graph of the histogram of all the ages of all the pennies in your class shown below to
describe the distribution.
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Age of Pennies in Years
140
Frequency
120
100
80
60
40
20
0
Age of Pennies (2010 = 0 years)
3. Estimate the mean and standard deviation of the distribution. Confirm these estimates by
actual computations.
Estimated mean: _______
Actual mean: __________
4. Take a sample of pennies of size 5 and find the mean age of your pennies. (remember, a
penny from the year 2010 is 0 years old, a penny from 2007 is 3 years old). Record your
sample below and then find the mean age.
Year of pennies: ______
Age of pennies: ______
______
______
______
______
______
______
______
______
sample’s mean age = ________
***Once you have found your mean, add the value to the class dot plot***
5. Copy the class dot plot in the space provided below:
6. Do you think the mean of the values in this histogram will be larger than, small than, or the
same as the one for the population of all pennies? Regardless of which you choose, try to
make an argument to support your choice.
____________________________________________________________________
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7. Now enter the class data (all of the means) into your calculator and determine the mean
and standard deviation of the sampling distribution. Which of the three options from
question 6 appears to be correct?
mean of sampling distribution = _____________
____________________________________________________________________
8. Repeat this experiment for size 10:
Penny ages: ____ ____ ____ ____ ____ ____ ____ ____ ____ ____
sample’s mean age = _______
9. Copy the class dot plot in the space provided below:
10. Repeat the experiment for size 20:
Penny ages: ____ ____ ____ ____ ____ ____ ____ ____ ____ ____
____ ____ ____ ____ ____ ____ ____ ____ ____ ____
sample’s mean age = _______
11. Copy the class dot plot in the space provided below:
12. Look at the four histograms/dot plots that your class constructed. What can you
say about the shape of the histogram as n increases?
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13. What can you say about the spread of the histograms as n increases?
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WRAP-UP
1. The three characteristics you examined in this activity (shape, center, and spread
of the sampling distribution) make up the central limit theorem. Without looking
in your textbook or notes, write a statement of what you think the central limit
theorem says.
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2. The distributions you constructed for 1, 5, 10, and 20 are called sampling
distributions of the sample mean. Sketch the sampling distribution of the sample
mean for samples of size 40. (just a sketch…use the same scale as before)