122
Supplement: Linear Combinations of Random Variables CLASS EXERCISES
CONTINUED: The graph of
x
Exercise 1.
Take a “Minitab die” and roll it 1000 times.
(
Calc > Random data > Integer; generate 1000
rows; min: 1; max: 6
)
Record the following statistics (
Stat > Basic Statistics > Display Descriptive Statistics
).
Round your
values to 1 decimal place.
(*)
The mean
x
of the 1000 rolls:
(*)
The median of the 1000 rolls:
The range of the 1000 rolls (if it’s not 5, I’d suggest playing the lottery tonight!):
The standard deviation of the 1000 rolls:
Plot the mean, median, and range of your rolls on the graphs at the front of the room.
Are there any special
patterns to the graphs?
Graphs:
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Exercise 2.
In this second exercise, we will investigate changes in the sampling distribution of the
sample mean
x
as we increase the size of the sample from 1 to 4 while sampling from a Uniform(0, 1)
population.
We will do this via simulation using Minitab.
1.
Use
Calc > Random Data > Uniform
to get 1000 values from the Uniform(0, 1) distribution by
specifying 1000 rows.
Store these values in column C1.
2.
Each row in C1 represents a sample of size 1 from the Uniform(0, 1) distribution.
Construct a histogram
of the values in C1.

After the histogram is drawn, right click on the histogram (of each) and select “Edit Bars”

Select the menu for “Binning;” Select “Interval Type” as Cutpoint

Select the “Interval Definition” as Midpoint/Cutpoint Positions and type in (or cut and paste) the
endpoints for C1 (space in between each):
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
Compute the mean and standard deviation of these values using Minitab's descriptive statistics command.
Mean of C1:
StdDev of C1:
3. Repeat step 1, but store the values in column C2.
4. The values in C1 and C2 for each row represent a
simple random sample of size 2
from the Uniform(0,
1) distribution.
Average these two values, (C1+C2)/2, and store the result in C3.
Use
Calc > Calculator
to
average and store the values in C3.
5.
Each row in column C3 represents a
sample mean
from a simple random sample of size 2.
Construct a
histogram of these means.
Note the shape of the distribution; it’s called a triangular distribution.
Also,
compute the mean and sample standard deviation of these values.
Mean of C3:
StdDev of C3:
Is the mean larger, smaller, or the same for C3 as compared to C1?
Is the standard deviation larger, smaller, or the same for C3 as compared to C1?
6.
Repeat step 1, but store the values in columns C5 and C6.
We want to enlarge the sample size to 3, then
4.
Compute the corresponding sample means, (C1 + C2 + C5)/3 and (C1 + C2 + C5 + C6)/4, then generate
the histogram and compute the means and sample standard deviations.
Shape of graph of C5?
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 Spring '08
 DeVasher
 Statistics, Normal Distribution, Standard Deviation, RoseHulman, Cutpoint, Olin elevator

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