Activity 1:
Using the TI-83 calculator to
Find Factorials, Binomial Coefficients, and Binomial Probabilities

A binomial experiment is defined by the following conditions:
1. The experiment is repeated for a fixed number of trials (n).
2. Trials are independent.
3. For each trial, there are two mutually exclusive outcomes: success or failure.
4. The probability of a success, p, is the same for each trial.
5. The random variable x counts the number of successful trials.

Binomial probability formula:
P( x) 
n!
 p x  (1  p)n x n Cx  p x  (1  p)n x   nx   p x  (1  p) n x
(n  x)! x!
where x is the number of successes in n trials, p is the probability of success in any
one trial, and (1-p) is the probability of failure in any one trial.

Mean and standard deviations for a binomial distribution
  np
  n * p *(1  p)
1) Evaluate factorials with the calculator:
Type number
Press MATH
Arrow left to PRB
Select 4:!
Press ENTER
Examples:
a) Find 10! = 3628800
b) Find
10!
= 120
3!* 7!
2) Evaluate binomial coefficients with the calculator:
Type n (number of trials)
Press MATH
Arrow left to PRB
Select 3:nCr
Type x (number of successes in n trials)
Press ENTER
Example:
Find 10 C3 = 120
1
3) Find binomial probabilities with the calculator
A) To find single probabilities: Use binompdf(n,p,x)
Press 2nd VARS[DISTR]
Select 0:binompdf(
Type n,p,x)
Press ENTER
Example:
For a binomial experiment with n = 7 and p = 0.8, find P(x = 3). Use the formula given on the
previous page to check the answer.
Binompdf (7, .8, 3) = 0.028672
B) To get a list of all the probabilities corresponding to x = 0, 1, 2, …., n:
Use binompdf(n,p) and scroll to the right to read the probabilities
Example:
For a binomial experiment with n = 4 and p = 1/6, find the probability distribution.
Binompdf(4, 1/6)
X P(X=x)
0 0.4823
1 0.3858
2 0.11574
3 0.01543
4 0.0078
C) To calculate cumulative probabilities from 0 to x, use binomcdf(n,p,x)
Press 2nd VARS[DISTR]
Select A:binomcdf(
Type n,p,x)
Press ENTER
Examples:
a) For a binomial experiment with n = 7 and p = 0.2, find the probability of at most 3
successes.
P(x ≤ 3) = binomcdf (7, .2, 3) = 0.9667
b) For a binomial experiment with n = 8 and p = 0.85, find the probability of at least 5
successes.
P(x ≥ 5) = binomcdf (8, .85, 4) = 0.02135
c) For a binomial experiment with n = 9 and p = 0.35, find the P (2 < x <6)
P(x = 3 or x = 4 or x = 5) = binomcdf(9, 0.35, 5) – binomcdf (9, 0.35, 2) = 0.6091
2
Activity 2:
Using the TI-83 calculator to
Find Normal Distribution Probabilities
Find Scores given Probabilities

Normal models are appropriate for distributions whose shapes are unimodal and roughly
symmetric.
To find probabilities (areas under the normal curve) we can use the tables provided in
any Statistics book. They are based on the z-scores. A z-score is the number of standard
deviations that a given score is above or below the mean.
z
x

. We’ll use
technology in this case.
4) Find normal distribution probabilities with the calculator.
Press 2nd VARS[DISTR]
Select 2:normalcdf(
Type lower bound, upper bound, mean, standard deviation)
Press ENTER
Example:
Assume that men’s weights are normally distributed with a mean of μ = 172 lb and a standard
deviation of σ = 29 lb. If one man is randomly selected, find the probability that his weight is
a) Less than 167 lb. (Use -1E99 for lower bound)
P(x < 167) = normalcdf (-1E99, 167, 172, 29) = 0.431556
b) More than 170 lb. (Use 1E99 for upper bound)
P(x > 170) = normalcdf (170, 1E99, 172, 29) = 0.52749
c) Between 170 and 175 lb.
P(170 < x < 175) = normalcdf (170, 175, 172, 29) 0.0
d) If 64 men are randomly selected, find the probability that they have a mean weight between
170 and 175 lb. (Here, according to the Central Limit Theorem you have to use the standard
deviation of the distribution of sample means which is  / n .)
P(170 < x bar < 175) = normalcdf (170, 175, 172, 29/sqrt64) = 0.505479
3
5) Find scores given areas (probabilities) under the normal curve.
Press 2nd VARS[DISTR]
Select 3:invNorm(
Type area to the left, mean, standard deviation)
Press ENTER
Example:
Find the first quartile of the distribution of the men’s weights from the previous problem
(μ = 172 lb, σ = 29 lb). That is, find the height that separates the lower 25% from the top 75%.
InvNorm(.25, 172, 29) = 152.4 lb
4
Activity 3
Using the TI83 calculator to
Construct Binomial Probability Distributions and Histograms
6) Use the calculator to construct the probability distribution and histogram for the binomial
experiment with n = 12 and p = .5
a) Store the possible values of the random variable, 0 – 12, into L1. Here is a way of doing this:
The instructions in the home screen of the calculator should read
Seq(x,x,0,12,1) →L1
Where to find this:
The Sequence feature is in the 2nd STAT[LIST] menu (arrow right to OPS)
The STOre key is in the first column of the calculator
L1 is the 2nd function of the key for number 1
b) Calculate the probabilities for each value of the random variable and store them into L2.
The instructions in the home screen of the calculator should read:
Binompdf(n,p)→L2
Where to find this:
Binompdf(…….. is in the 2nd VARS[DISTR] menu
c) Explore the editor of the calculator to make sure you have the probability distribution in lists
1 and 2 of the calculator.
How to do this:
Press 2nd STAT, select 1:Edit
d) Construct the probability histogram for the experiment. Use plot 1 to graph the data stored
into L1 and L2.
How to do this:
Set up the histogram in the 2nd Y=[STAT PLOT] window
Use the WINDOW [-3.5, 15.5, 1] for x and [-.3, .5, 1] for y
Press GRAPH
Press TRACE and scroll right to observe the probabilities
e) Describe the shape of the distribution.
Symmetric
Some of the probabilities for x = 0, 1, 2, … are here
0.00024, 0.00293, 0.01611, 0.05371, 0.12085, 0.19336, 0.22559, …..
5
Activity 4
Using the TI83 calculator to
Find the Mean and Standard Deviation of a Binomial Distribution
7) Use the calculator to find the mean and the standard deviation for the binomial probability
distribution from problem 6 in which n = 12 and p = .5
Once you have the values of the random variable in List 1 and the probabilities in List 2 (which is what
you did on the previous page), do the following
The instruction in the home screen of the calculator should read
1-Var Stats L1,L2
Here are the steps to accomplish that
Press STAT
Arrow to CALC
Select 1:1-Var Stats
Indicate location of data: L1,L2
Press ENTER
You can check your answers with the formulas
  n* p
  n * p * (1  p)
μ = 6, σ= 1.732
6
Activity 5
Using the TI83 calculator to
Graph Normal Distributions
Explore the Normal Approximation to a Binomial Distribution
8) Graph the normal distribution that has the same mean and standard deviation as the binomial
distribution from problem 7
a) Here is what we just found on the last page for the binomial experiment in which n = 12 and p = 0.5.
  n * p  12 * 0.5  6
  n * p * (1  p)  12 * 0.5* 0.5  3
b) Graph the normal distribution with mean 6 and standard deviation 3 .
Press Y=
Y1= normalpdf(x, mean, standard deviation)
(From the 2nd VARS[DISTR] menu)
Press GRAPH
(Keep Plot1 ON, and use the same window as in Activity 3)
c) In your opinion, how well does the normal distribution approximate the binomial distribution for
n = 12 and p = 0.5?

Normal Distributions as Approximation to Binomial Distributions
If np  5 and n(1  p)  5 , then the binomial random variable has a probability
distribution that can be approximated by a normal distribution with the mean and
standard deviation given as
  n* p
  n * p * (1  p)
7
Activity 6
Probability Distribution for the Binomial Experiment:
Rolling a fair die 5 times and recording the number of ones obtained
9) This is similar to what you did on ACTIVITY 3. If you do not want to repeat it, it is fine, just
take a look at the shape of this distribution because the next activity will simulate this
experiment.
Probability Distribution
(1) To obtain the histogram:
L1: number of ones obtained
Seq(x,x,0,5,1)→L1
nd
In the 2 STAT[LIST], OPS menu
L2: probabilities
binompdf(5,1/6)→L2
in the 2nd VARS[DISTR] menu
Press 2nd Y=, turn a plot ON, select the histogram
and indicate the location of the data L1,L2
Press WINDOW
Xmin=0, xmax=6, x-scl=1
Ymin=-.2, ymax=.5
Press GRAPH
x
Just a “side” comment to stress what we did in the previous activity 5
In your opinion, how well does the normal distribution approximate the binomial distribution for this
experiment in which n = 5 and p = 1/6?
Notice that
  np  5(1/ 6)  .833
  npq  5(1/ 6)(5/ 6)  .833
You may also want to graph the normal distribution with these parameters, as we did in activity 5:
Y1 = Normalpdf(x,5*1/6,sqrt(5*(1/6)*(5/6))
(from the 2nd VARS[DISTR] menu)
8
Activity 7
Simulation of the Experiment of
Rolling a fair die 5 times and recording the number of ones obtained
Constructing the Probability Distribution for the Experiment
10) Simulate rolling a die 5 times. Record the number of ONES obtained.
The instruction in the home screen of the calculator should read
randInt(1,6,5)
(this generates 5 random integers from 1 to 6)
Where to find it?
Press MATH, arrow to PRB, select 5:randInt(
Press ENTER
a) Count the number of ONES obtained and tally on the table.
Do the process 10 times (Press ENTER 10 times).
Record your frequencies on the board. (We will not do this in the workshop)
Use the class’ results to construct the probability distribution of the experiment. (We will not do this in the
workshop)
Number of
ONES
0
1
2
3
4
5
Tally
Your Frequency
Class’ Frequency
Probability
b) Compare the results with the “theoretical” probability distribution from the previous page. How can
you get the experimental probabilities to be closer to the theoretical probabilities?
c) Which values of the random variable are more likely? Use common sense to answer, then look at the
probabilities and see if they agree with your prediction.
d) Find the experimental probability of obtaining zero ONES. Compare with the theoretical results.
e) Refresh your memory on how to calculate the theoretical probability of obtaining zero ONES when
rolling a die 5 times.
1. Use the formula.
2. Use a calculator feature.
9
Activity 8
A Different Simulation of the Experiment of
Rolling a fair die 5 times and recording the number of ones obtained
Sampling from a Binomial Distribution
Sampling Error
11) Let’s try a different simulation now.
We can think that we select samples of size n from this “always perfect theoretical” binomial
distribution for the experiment, which is shown in activity 6 (page 8). Each member of the sample
represents the number of ONES obtained when we roll the die 5 times.
We can simulate the sampling process of selecting numbers from this distribution by using
randBin(number of trials, probability of success, sample size)
We are going to store the numbers in L1, set up a histogram, set up the correct window and observe the
shape of the distribution.
Here is how to accomplish the selection of TEN numbers from this distribution:
randBin(5,1/6,10)→L1
(in the MATH, PRB menu)
Set up Stat Plot to histogram (2nd Y=[STAT PLOT])
Open window: x-min = 0, x-max = 6, x-scl = 1, y-min = -5, y-max = adjust as
appropriate
Press GRAPH
To repeat the process again, 2nd QUIT to go to the HOME SCREEN
Press 2nd ENTER to capture the last ENTRY
Press ENTER again
Press GRAPH and observe the shape of the new sample.
Try different sample sizes and then think on the LAW OF LARGE NUMBERS
Samples of Size n = 10
Parent Population
x
x
Samples of Size n = 30
x
x
10
Activity 9
Sampling from a Binomial Distribution
Observing the Distribution of Sample Means
12) Sampling from the parent population of the experiment of “rolling a die 5 times and counting
the number of ONES obtained”
a) Select a sample of size 10. Find the mean of the number of ONES obtained.
Repeat the process many times and observe the variability of the means.
The instruction in the home screen of the calculator should read
Sum(randBin(5,1/6,10))/10
OR
mean(randBin(5,1/6,10)
Here are the steps to accomplish that
2nd STAT[LIST], arrow to MATH, select 5:sum(
Press MATH, arrow to PRB, select 7:randBin(
Press ENTER many times
or 3:mean(...
b) Now select a sample of size 30. Repeat the process many times and observe the variability of the
means.
c) Now select a sample of size 60. Repeat the process many times and observe the variability of the
means.
d) Comment on the variability of the distribution of sample means as n increases
11
Activity 10
Using Applets to Explore
Shapes of Binomial Distributions
13) Binomial Demonstrations
Explore how the values of n and p affect the shape of a Binomial Distribution
Access my web page: http://montgomerycollege.edu/~maronne/
Click on Statistics Workshop
Click on Applets
Click on Binomial Demonstrations
Scroll down and, within the Statistical Applets section,
Click on Binomial Demonstration (it is on the second column)
CASE I) Let p = 0.5.
a) Move the slider to change the sample size n. Let the slider run from the smallest value of n to the
largest and observe the shape of the distribution. Try this many times and comment on the shape of the
distribution. What do you observe?
b) Complete the following table:
n=
p = 0.5 n*p = ....... n*p = .......
10
20
40
........
Comment on the shape of the distribution
c) With p = 0.5, is there any value of n for which the shape of the distribution is other than bell shaped?
12
CASE II) Let p = 0.2.
a) Move the slider to change the sample size n. Let the slider run from the smallest value of n to the
largest and observe the shape of the distribution. Try this many times and comment on the shape of the
distribution. What do you observe?
b) Complete the following table:
N = p = 0.2 n*p = ....... n*p = .......
10
20
40
Comment on the shape of the distribution
c) With p = 0.2, what value of n starts producing an approximately bell shaped distribution? Find np
and nq for this value of n.
CASE III) Let p = 0.7.
a) Move the slider to change the sample size n. Let the slider run from the smallest value of n to the
largest and observe the shape of the distribution. Try this many times and comment on the shape of the
distribution. What do you observe?
b) Complete the following table with your observations:
N = p = 0.7 n*p = ....... n*p = .......
10
20
40
Comment on the shape of the distribution
c) With p = 0.7, what value of n starts producing an approximately bell shaped distribution? Find np
and nq for this value of n.
14) Summarize your conclusions about the shape of a binomial distribution based on the values
of n and p. Refer to Cases I-c, II-c, III-c.
13
Activity 11
Using Applets to Explore
Normal Distributions
15) Explore how the CENTER and the SPREAD of the normal distribution are affected as the
values of the mean and the standard deviation change
Access my web page: http://montgomerycollege.edu/~maronne/
Click on Statistics Workshop
Click on Applets
Click on Normal Distributions 1

First we’ll try changing the value of the mean while keeping the same value for the
standard deviation
Change the value of the mean to 1, and click on DRAW
WHAT DO YOU OBSERVE? What is changing: the center, the spread, or both?
Try different values for the mean: 1.5, 2, 0.5, -1, etc.
WRITE YOUR CONCLUSIONS HERE:
 Now fix the mean to zero and change the standard deviation
Change the value of the standard deviation to 1.5, and click on DRAW
WHAT DO YOU OBSERVE?
Try the same with different values for the standard deviation: 2, 0.5, 0.1, etc.
WRITE YOUR CONCLUSIONS HERE:
 Now change both, the mean and the standard deviation
WHAT DO YOU OBSERVE?

Now answer the following:
Can the mean be negative?
Can the standard deviation be negative?
If we only change the mean, is the “height” of the normal curve changing?
What produces a change in the “height” of the normal curve? Why?
The mean determines the __________ of the distribution.
The standard deviation determines the _________ of the distribution.
14
Activity 12
Using Applets to Explore
Normal Distributions
16) This is a different applet that will explore the same concepts explored in Activity 9.
Access my web page: http://montgomerycollege.edu/~maronne/
Click on Statistics Workshop
Click on Applets
Click on Normal Distributions 2
Scroll down and click on Flash Demo (It is the 11th link on the third column)
Scroll Down and click on Normal Distribution
“Grab” the red square on the μ scale to change the value of the mean. Try different values.
What do you observe?
“Grab” one of the red squares on the σ scale to change the value of the standard deviation. Try
different values. What do you observe?
Click the “fill” box. The green area in this applet represents the area under the normal curve within
one, two, three (circle one)
standard deviations from the mean.
In a normal distribution, no matter what the values of the mean and the standard deviation are, the
value of the “green” area shown in this applet is always _________.
About __________% of the scores are within 1 standard deviation from the mean.
15
Activity 13
Using Applets to Explore
Normal Approximation to Binomial Distributions
17) When can we use the Normal distribution to approximate a Binomial distribution?
Access my web page: http://montgomerycollege.edu/~maronne/
Click on Statistics Workshop
Click on Applets
Click on Normal Approximation to Binomial
Scroll down and click on Normal App to Bin
Read instructions. Change values of n and p, observe and summarize what you learned.

Normal Distributions as Approximation to Binomial Distributions
If np  5 and n(1  p)  5 , then the binomial random variable has a probability
distribution that can be approximated by a normal distribution with the mean and
standard deviation given as
  n* p
  n * p * (1  p)
16
Activity 14
Using Applets to Explore
Sampling Distributions of the Mean
Central Limit Theorem
18) Explore the sampling distribution of the mean.
Access my web page: http://montgomerycollege.edu/~maronne/
Click on Statistics Workshop
Click on Applets
Click on Sampling Distributions
Click the Begin button on the left, and read the instructions given below.
You are basically given a PARENT POPULATION which shape is shown on the top of the
screen.
 Change the shape of the “parent population” by clicking on the shown graph, or by selecting
different shapes on the drop down window shown to the right of the graph.
 Explore different shapes for the “parent population”.
 The CUSTOM design can be done by clicking on the graph.
 Try all of the above before moving ahead.
Now we are going to simulate the process of selecting 5 numbers from a NORMAL population,
computing the mean of the five numbers, and plotting the mean of the five numbers
 Select NORMAL for the shape of the “parent population”.
 On the third group of buttons on the right of the screen, select Mean and N=5
 Click on Animated Sample (second group of buttons on the right)
 The second graph will show the 5 numbers selected from the parent population and the third
graph will show the mean of those five observations.
 Click on Animated Sample again, and again, and again….
As we repeat this process over and over again, we are creating a new distribution: the
distribution of sample means, which shows in blue on the third graph.
Now that you see how it works, you can speed things up by taking 5, 1000, or 10,000 samples at a
time.
 Click on the corresponding buttons on the right of the screen.
Observe the distribution of sample means (the blue histogram). What is the shape of that
distribution? Answer the following:
a) The parent population’s shape is:
b) The sample size is:
c) The shape of the distribution of sample means is:
17
19) Now we’ll start a new simulation with a larger sample size:
 Click on the Clear Lower 3 button to start the simulation again.
 Change to N = 25
 Click on Animated Sample and observe.
Explain the meaning of what just happened:
Speed up the process by selecting more than one sample at a time, observe the distribution of
sample means (the blue histogram), and describe its shape:
Answer the following:
a) The parent population’s shape is:
b) The sample size is:
c) The shape of the distribution of sample means is:
20) Now start a new simulation with a different shape for the parent population.
Try at least two sample sizes, a small, and a large.
Answer the following:
a) The parent population’s shape is:
b) The sample size is:
c) The shape of the distribution of sample means is:
21) After “playing” with different selections of shapes for parent populations and sample sizes,
what do you conclude about the shape of the distribution of sample means in relation to the
sample size?
18
Activity 15
Using Applets to Explore
Sampling Distributions of the Mean
Central Limit Theorem
22) Here we have another applet to explore the sampling distribution of the mean.
Access my web page: http://montgomerycollege.edu/~maronne/
Click on Statistics Workshop
Click on Applets
Click on Central Limit Theorem
Scroll down and click on Flash Demo (It is the 11th link on the third column)
Scroll down and click on Mean and the Median
a) First move up and down the slider on the left, shown above N. This will select the sample size.
Now check and “un-check” the square next to Parent Distribution and observe the effect.
Now check and “un-check” the square next to Theoretical mean and observe the effect.
Now click on Skew Distribution, on the top-right to observe the effect
Try all of the above before moving ahead.
b) To start the simulation, you want to check only the squares next to Fast, Parent Distribution,
Theoretical mean, History of means
Also work with a Symmetric Distribution (top-right)
Click on New Start
Click on New Sample
Click off History of medians if it clicks on by itself (the medians are in green, and it will be confusing
to watch the history of medians along with the history of means)
Click on New Sample over and over again. Explain what is happening. What are the blue lines you
see?
What is the red line you see?
When you understand what is going on, click on Run, and let it run for a while.
You will see the distribution of sample means being formed on the bottom.
If we repeat the process over and over again for a large number of times, we expect the shape of the
distribution of sample means to be ……………………..
19
c) Explore the sampling distribution of the mean from a skewed parent population.
In the same applet, select a skewed parent population. (Top right)
Explore the sampling distribution of the mean for a small sample (n = 5) and then for a large sample
(n = 35). Comment on the shape of the distribution of the sample means.
For n = 5, the shape of the distribution of sample means is……………………….
For n = 35, the shape of the distribution of sample means is……………………….
d) In your own words, what does the central limit theorem say?
20
Activity 16
Using Applets to Explore
Variability of the Distribution of Sample Means (DSM)
23) Explore the variability of the DSM
Access my web page: http://montgomerycollege.edu/~maronne/
Click on Statistics Workshop
Click on Applets
Click on Sample Size vs. Variability of DSM
Instructions
In this applet we are drawing samples from a population that has a mean of 100. We’ll select a certain
number for Sample size by moving the slider and then we’ll click on the Sample button and the mean
of the sample will be plotted. We’ll repeat this process many times for each of the sample sizes
selected.
Select the sample sizes indicated in the table given below and after clicking the Sample button many
times, comment on the mean and standard deviation of the new distribution of sample means.
Sample Size
Mean of sample means
Variability of the sample
means compared to the
variability of the original
distribution
1
4
9
16
25
Write a paragraph with your observations.
21
Sharing results from my classes
Sampling from a Binomial Distribution
Graphing the Distribution of Sample Means
For Samples of Size n
The Central Limit Theorem
Here are the graphs obtained with the data collected in my classes
Distribution of Sample Means for Samples of Size 10
x
 x  .864, x  .307
( /
(For the distribution obtained in class)
n  .83333333 /
10  .263 )
Distribution of Sample Means for Samples of Size 30
x
 x  .835, x  .159 (For the distribution obtained in class)
(  / n  .8333333/ 30  .152 )
The Central Limit Theorem
And the Distribution of Sample Means
As the sample size increases, the sampling distribution of sample means approaches a normal
distribution with the following mean and standard deviation
x  
x 

n
22
Summary
Sampling from a Binomial Distribution
Central Limit Theorem
Original Distribution
Samples of Size 30
(n = 30)
x
Probability Distribution
L1: outcome
L2: probabilities
Distribution of
Sample Means for
Samples of Size 30
x1  1.1
x2  .766667
x
x3  .566667
x
μ = np = .8333333,
σ = sqrt(npq) = .833333
The Central Limit Theorem
And the Distribution of Sample Means
As the sample size increases, the sampling
distribution of sample means approaches a
normal distribution with the following mean
and standard deviation
Distribution of x , n = 10
x
Distribution of x , n = 30
x  
x 

n
x
23
Summary
Normal Approximation to Binomial Distributions
When can we use the Normal distribution to approximate a Binomial distribution?

Normal Distributions as Approximation to Binomial Distributions
If np  5 and n(1  p)  5 , then the binomial random variable has a probability
distribution that can be approximated by a normal distribution with the mean and
standard deviation given as
  n* p
  n * p * (1  p)
Case I: n = 12, p = 0.5
Case II: n = 12, p = 0.2 (p < 0.5)
Case III: n = 12, p = 0.9 (p > 0.5)
24