Download Investigating a Distribution of Sample Means

Distribution of the Sample Mean
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Investigating a Distribution of Sample Means
The goal of this activity is to use a graphics calculator investigate a distribution sample means
from a collection of pocket change through a simulation.
Earlier in the term, we brought in a sample of pennies found in our pocket change. The years
when the pennies were minted were entered into a list called LCOIN. The purpose of this activity
is to determine the nature of the distribution of the means of samples of 4 pennies.
Step 1: Determine the distribution of the population of pennies in our collection.
1. Graph a histogram of the list LCOIN in STAT PLOT 1 using the window Xmin = 1950, Xmax
= 2010, and Xscl = 10 and an appropriate Ymin, Ymax, and Yscl.
2. Determine the mean µ and standard deviation σ of the data. Remember, this is a population
of pennies in our collection.
3. Discuss the distribution, including the center, spread, and shape.
To following commands will help build the process of obtaining a distribution of sample means
of the years the pennies in our collection were minted
To make a random selection of a penny from our collection, enter the following command:
LCOIN(randInt(1,dim(LCOIN)))
What value did you get? In what year was the penny minted?
To make a random selection of 4 pennies from our collection, enter the following command:
seq(LCOIN(randInt(1, dim(LCOIN))),X,1,4,1)
What values did you get? In what years were the pennies minted? What is the mean of these
years?
To determine the mean of a random selection of 4 pennies from our collection, enter the
following command:
mean(seq(LCOIN(randInt(1, dim(LCOIN))),X,1,4,1))
What is the mean of the years the pennies were minted in this (unknown) sample?
Adapted from Barrett (1999)
The Mathematics Teacher
Distribution of the Sample Mean
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Step 2: Store 100 sample means in L1.
4. Initialize a counter variable. The following command will set the variable C to 0:
0C
5. Clear L1. You may either do this in the STAT EDIT menu or execute the following
command:
ClrList L1
6. Enter the sample means into L1. The following command includes three instructions
combined with a colon (an ALPHA character above the decimal point). The first instruction
updates the counter variable C. The second instruction finds the mean year of a random
sample of 4 pennies taken from the list LCOIN and stores it in L1 in the position determined
by the counter C. The third instruction displays the value of C, which is the only instruction
seen on the HomeScreen.
C+1C:mean(seq(LCOIN(randInt(1,dim(LCOIN))),X,1,4,1))L1(C):C
Repeat the command for a total of 100 times.
Step 3: Determine the distribution of the sample means of pennies in our collection.
7. Graph a histogram of the list L1 in STAT PLOT 2 using the window [1950,2010,2] and
[-10,30,5]. Be sure to turn off STAT PLOT 1.
8. Determine the mean and standard deviation of the sample means. Remember, this is a
sample of sample means.
9. Discuss the distribution, including the center, spread, and shape.
Step 4: Verify the Central Limit Theorem.
10. Verify the distribution of the sample means is normally distributed. The critical value for the
correlation coefficient for a sample size of 100 is .987 (Looney & Gulledge, 1985).
11. Verify the mean and standard deviation of the distribution of the sample means is
approximately equal to the predicted values.
x x ≈ µ and s x ≈
σ
n
Adapted from Barrett (1999)
The Mathematics Teacher