Download Project 3 – The Central Limit Theorem

Project 3 – The Central Limit Theorem - 20 points
Due: Thursday, February 13
Name:_______________________
No Late Projects Will Be Accepted
PART 1: Create and Examine a Theoretical Population
(a) Let X = the number of dots facing up
when one fair die is rolled. Find the
probability distribution for the random
variable, X, and summarize it in the
table.
X
P(X)
(b) Neatly construct the probability histogram. Label the axes.
(c) Describe the probability histogram (symmetry, shape, center, normal).
(d) Find the mean of this discrete random variable, X. (Reminder: This is asking you to find the
mean of a random variable…not a sample mean.) Show your calculation.
(e) Find the standard deviation of the random variable, X. (Reminder: This is asking you to find
the standard deviation of a random variable…not a sample standard deviation.) Keep
EXACT fractions throughout the calculation. Round to three digits after the decimal for your
final answer only. Show your calculation.
PART 2: SIMULATION
(a) Now you will roll two dice, so it is as if you are taking a sample of size 2 from the
theoretical population you described in PART 1, and let X = the mean of the two numbers
showing. We know X is a random variable. Think about the possible values that X can be.
(Hint: There are 11 possible X values when you roll two dice.) List them in the table
below.
Now you will simulate taking 50 samples of size n = 2 from the population in PART 1 by
actually rolling the two dice 50 times. Every time you roll the two dice, find X , and make a
tally mark in the table to show how often you observed each X .
NOTE: You can do this dice rolling at home using two dice of your own or on the Internet at
a site that rolls the dice for you. The following site will roll dice for you. Make sure you set
the number of sides as 6 and the number of dice as 2. You need to calculate X after each roll
and tally the results in the table.
http://dice.virtuworld.net/
X
Observed Frequency Tallies
(b) Neatly construct a frequency histogram using the simulation results from part (a). Label the
axes.
(c) Examine the frequency histogram of X you created in PART 2(b). What type of distribution
does it appear to be (symmetric or not, normal or not, center)?
(d) In PART 2(a) and (b) you used data to explore the distribution of the sample means of 50
samples of size 2 taken from the population in PART 1 but now let’s construct the theoretical
probability distribution so we can get an accurate mean and standard deviation for the
distribution of X . To determine the theoretical probabilities, it will be helpful to list all
possible equally likely outcomes in a table. In the following table, fill in X for each possible
pair of dice.
X
1
2
3
Die 2
4
5
6
1
2
Die 1 3
4
5
6
Now, use the fact that there are 36 equally likely outcomes to calculate the probability of each
of the 11 different values of X . Insert all this information in the following table. Give the
value of X along with the exact probability X takes on a particular value.
Probability Distribution of X :
X
P X
 
(e) Construct the probability histogram for X . Label the axes.
What type of distribution is this (symmetric or not, normal or not, center)?
(f) Find the mean of the random variable X . Show your calculation.
(g) Find the standard deviation of the random variable X . Do not round probabilities in the
calculation. Keep three digits after the decimal in your final answer. Show your calculation.
PART 3: Putting It All Together
(a) Compare the mean from PART 1(d) to the mean from PART 2(f). What do you notice?
(b) Compare the standard deviation from PART 1(e) and to the standard deviation from PART 2(g).
Which standard deviation is smaller?
(c) If you take the standard deviation from PART 1(e) and divide it by 2 (the square root of the
sample size) how does it compare to the standard deviation in PART 2(g)? _______________
So now finish this sentence: If we know the standard deviation of a population and we take lots
of samples of size 2, then we can predict the standard deviation of X by
calculating_________________________________________________________.
(d) Carefully read about the Central Limit Theorem on page 161. State the Central Limit Theorem
in your own words and be sure to talk about the mean of X and standard deviation of X .
(Ignore references to Sn.)
(e) Did your results from PART 3(a) and 3(c) support the Central Limit Theorem? (They should. If
they did not then go figure out why and fix it.) Be specific involving the means and standard
deviations.
(f) If we would take samples of size n = 30 from the population in PART 1 and find X every time,
then we would expect the mean of the random variable X to be __________ and the standard
deviation of random variable X to be _________.
We would expect the distribution of X to be a ____________________ distribution.