Find Factorials, Binomial Coefficients, and Binomial Probabilities

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:

n! x n- x P( x )=贩 p (1 - p ) =C贩 px(1 - p ) n- x =n贩p x(1 - p ) n- x (n- x )! 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 m = np s =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

10! b) Find = 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 10C 3 = 120 3) Find binomial probabilities with the calculator

1 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 x - m deviations that a given score is above or below the mean. z = . We’ll use s 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 s / 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, …..

Activity 4 5 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 m = n* p μ = 6, σ= 1.732 s =n* p *(1 - p )

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.

m =n* p = 12*0.5 = 6 s =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 m = n* p s =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 Press 2nd Y=, turn a plot ON, select the histogram Seq(x,x,0,5,1)→L1 and indicate the location of the data L1,L2 In the 2nd STAT[LIST], OPS menu L2: probabilities Press WINDOW binompdf(5,1/6)→L2 Xmin=0, xmax=6, x-scl=1 in the 2nd VARS[DISTR] menu Ymin=-.2, ymax=.5 Press GRAPH

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 m =np =5(1/ 6) = .833 s =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 Tally Your Frequency Class’ Frequency Probability ONES 0 1 2 3 4 5 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

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( or 3:mean(... Press MATH, arrow to PRB, select 7:randBin( Press ENTER many times

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 = ...... Comment on the shape of the distribution 10 20 40 ...... 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 = ...... Comment on the shape of the distribution 10 20 40 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 = ...... Comment on the shape of the distribution 10 20 40 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.

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:

19) Now we’ll start a new simulation with a larger sample size:

17  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

mx=.864, s x = .307 (For the distribution obtained in class) (s /n = .83333333 / 10 = .263 )

Distribution of Sample Means for Samples of Size 30

mx=.835, s x = .159 (For the distribution obtained in class) (s /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

mx = m s s = x n

22 Summary Sampling from a Binomial Distribution Central Limit Theorem

Original Distribution Samples of Size 30 Distribution of (n = 30) Sample Means for Samples of Size 30

Probability Distribution L1: outcome L2: probabilities x

μ = np = .8333333, σ = sqrt(npq) = .833333

Distribution of x , n = 10 The Central Limit Theorem And the Distribution of Sample Means

As the sample size increases, the sampling distribution of sample means approaches a x normal distribution with the following mean and standard deviation Distribution of x , n = 30

mx = m s s = x x n

23 Summary Normal Approximation to Binomial Distributions

When can we use the Normal distribution to approximate a Binomial distribution?

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)