# Distinguishing Between Binomial, Hypergeometric and Negative Binomial Distributions

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 Distinguishing Between Binomial, Hypergeometric and Negative Binomial Distributions Jacqueline Wroughton Northern Kentucky University Tarah Cole Northern Kentucky University Journal of Statistics Education Volume 21, Number 1 (2013), Copyright 2013 by Jacqueline Wroughton and Tarah Cole all rights reserved. This text may be freely shared among individuals, but it may not be republished in any medium without express written consent from the authors and advance notification of the editor. Key Words: Active Learning; Binomial; Hypergeometric; Negative Binomial; Project. Abstract Recognizing the differences between three discrete distributions (Binomial, Hypergeometric and Negative Binomial) can be challenging for students. We present an activity designed to help students differentiate among these distributions. In addition, we present assessment results in the form of pre- and post-tests that were designed to assess the effectiveness of the activity. Pilot study results show promise that the activity may help students recognize the differences in these three distributions. 1. Introduction Recently, a great deal of research has focused on active learning and hands-on activities in undergraduate statistics courses (e.g., Ledolter 1995; Chance 1997; delmas, Garfield and Chance 1999; Pfaff and Weinberg 2009). Most of the activities presented have been designed for the algebra-based introductory statistics class. However, this does not mean that students in the calculus-based introductory statistics class could not also benefit from the introduction of activities designed to improve understanding of certain statistical concepts. In addition to the complexity and level of the topics in the calculus-based course, there are issues in this course which are not completely attributed to the content, but also to the make-up of the student population. Many students who take a calculus-based course are confident in their mathematical ability, unlike many in the algebra-based course. This may pose a different kind of issue as students who are confident that they know what they are doing don t show much of their work 1

3 1. To simplify the implementation of probability calculations by utilizing their probability mass functions (pmfs) and to understand the general form of a pmf. 2. To determine and utilize their specific expected value and variance. 3. To learn about the concept of modeling outcomes of a given situation through demonstration. In order to understand how students may confuse three of these distributions, we will go through each of the three distributions characteristics in more detail. 2.1 Binomial Distribution When the Binomial Distribution is introduced, it is often done so by a list of conditions that must be satisfied. These five conditions (adapted from Wackerly, Mendenhall and Scheaffer 2008) are: 1. There is a fixed number, n, of identical trials. 2. For each trial, there are only two possible outcomes (success/failure). 3. The probability of success, p, remains the same for each trial. 4. The trials are independent of each other. 5. The random variable Y = the number of successes observed for the n trials. To determine if an experiment is Binomial, one just needs to examine whether or not each of the characteristics listed above is met. If the conditions of the Binomial Distribution are satisfied, then the following pmf can be used: n y n y f ( y; n, p) p (1 p) y 0,1, 2,..., n. y In addition, the expected value and variance can be utilized: 2.2 Hypergeometric Distribution E( Y ) np Var( Y ) np(1 p). The Hypergeometric Distribution arises when sampling is performed from a finite population without replacement thus making trials dependent on each other. However, when the Hypergeometric Distribution is introduced, there is often a comparison made to the Binomial Distribution. More specifically, it is said that if n is small relative to the population size, N, then (assuming all other conditions are met) Y could be approximated by a Binomial Distribution. This case is made due to the fact that not replacing the item has a negligible effect on the conditional p. However, when this is not the case, the independence condition is no longer met and the Binomial Distribution will no longer do an efficient job at an approximation since not replacing the item will have an effect on the conditional p. Consequently, the Hypergeometric Distribution should be used instead. So, when one compares the Hypergeometric to the Binomial conditions, one will see that conditions 1, 2, 3, and 5 still hold whereas condition 4 (given in Section 2.1), independence, no longer holds. 3

4 When the Hypergeometric Distribution is of interest, the following pmf can be used: r N r y n y f ( y; N, n, r ) y [max( 0, n ( N r )), min( r, n)]. N n Where r represents the number of success items out of the N total items. In addition, the expected value and variance can be utilized: nr ( N n) nr r E ( Y ) Var ( Y ) 1. N ( N 1) N N Comparing the pmf of the Binomial Distribution to that of the Hypergeometric Distribution, one can see that they are different due to the with replacement aspect of the Binomial Distribution compared to the without replacement aspect of the Hypergeometric Distribution. In addition, the support of y looks quite different between the two. This again is due to with replacement vs. without replacement aspect between the two distributions. For example, when drawing cards from a standard 52-card deck, suppose we are interested in Y = # of clubs drawn out of n. If n = 15, the most clubs we could draw is 13. If n = 45, the fewest clubs we could draw is 6. When one compares the expected value and variance of the two distributions, they appear to be very different. However, this is not really the case. If one were to see that r/n (of the Hypergeometric Distribution) is similar to p (of the Binomial Distribution), the expected values are the same and the variances are only different by the factor of (N-n)/(N-1), where the variances are identical in n=1; the variance of the Hypergeometric is smaller for n >1. This relates back to the idea that the Hypergeometric Distribution is used when the sample size, n, is no longer small in relation to the population size, N. However, when n is small in relation to N, this factor is negligible (and thus a Binomial Distribution may be appropriate as an approximation). 2.3 Negative Binomial Distribution When the Negative Binomial Distribution is introduced, it is often compared (and contrasted) to the Binomial Distribution. It has some of the same characteristics (conditions) as the Binomial Distribution, but has two distinct differences: The value of n (the number of trials) is no longer a fixed value and the random variable Y is defined differently; here Y is the number of trials needed to obtain r successes. That is, when looking at the conditions needed for a Binomial Distribution, conditions 2, 3, and 4 still hold whereas conditions 1 and 5 (given in Section 2.1) no longer hold due to the fact that the random variable and n basically change places. The number of successes, r, now becomes fixed and the number of trials, n, becomes the random variable. When the Negative Binomial Distribution is of interest, the following pmf can be used: y 1 r y r f ( y, r, p) p (1 p) y r, r 1,.... r 1 4

5 In addition, the expected value and variance can be utilized: E Y ) r p r(1 p) Var( Y ). p ( 2 Comparing the pmf of the Negative Binomial Distribution to that of the Binomial Distribution, one can see that they look pretty similar in construction, but the placement of the y is different (and of course, the number of combinations is slightly smaller). The support of y is quite different due to how the random variable, Y, is defined. In order to achieve r successes, one must have at least r trials. The maximum number of trials needed is not known. 2.4 Activity Motivation Looking at sections as the instructor, it appears that the differences in these three distributions are easy to discern; it seems as though one should just go through these five conditions that are needed for the Binomial Distribution to be met and figure out which ones hold and which ones don t hold. However, after having taught this class for multiple semesters, it appeared that the students had a tough time determining which of these distributions a situation followed, and why it would follow that distribution. More specifically, it appeared that the students had a tendency to believe the distribution would be Binomial more than any other of these three distributions. Thus, we felt that there was a need for the students to discover what it meant for independence to be violated, as well as seeing when n is no longer a fixed value. A hands-on activity was believed to be a good method because the impact of student discovery seems to trump the best teacher explanations. 3. The Activity In order to make scenarios similar and the situation straightforward, the design of this activity only requires a standard 52-card deck. Although the set-up is very straightforward, one could adapt what the cards represent to make the activity mimic a real-life application. For example, one could design a deck of cards where each red card represented a republican vote and each black card represented a democratic vote for a given election scenario. Another example might be to have red cards represent defective items and black cards represent satisfactory items where the students could be part of the quality control component of the company. (See Appendix A for the Activity Handout) For the activity, the students encounter three distinct scenarios (one for each distribution: Negative Binomial, Binomial, Hypergeometric respectively) and for each scenario the student is told how to shuffle and replace, as well as how the random variable, Y, is defined. This process and definition is what make the distinction among the three scenarios. Below we describe each scenario in more detail. In scenario one, the student is asked to draw a single card, then replace the card, shuffle, and repeat this process until two hearts have been obtained. The random variable, Y, is defined as the number of draws to achieve two hearts. Due to the fact that the card is replaced (and cards shuffled), independence is achieved. However, since the student does not know how many times that they will have to draw, they only know when to stop by achieving the second heart, the 5

6 number of draws is the random variable, thus making scenario one the Negative Binomial Distribution. In scenario two, the student is asked to draw a single card, then replace the card, shuffle, and repeat this process five times. The random variable, Y, is defined as the number of hearts drawn in the five draws. Due to the fact that the card is replaced (and cards shuffled), independence is achieved. The student knows that they will stop after drawing a total of five cards. Thus, this scenario meets all five conditions to follow the Binomial Distribution. In scenario three, the student is asked to draw five cards from the deck all at once. The random variable, Y, is defined as the number of hearts drawn in the five cards. Due to the fact that all five cards are drawn at once, independence no longer holds. The student knows how many cards they will draw. Thus, this scenario follows the Hypergeometric Distribution. In order for the activity to achieve enough data for demonstration of other values of interest, students are asked to go through each scenario multiple times (5-10). While students wait for the rest of the class to obtain their data, they can work on calculating the expected value and standard deviation of Y for each scenario and compute a theoretical probability using the appropriate pmf. After all students have obtained their data, they are also asked to compare the theoretical probabilities that they calculated to the empirical probabilities obtained by the class. Time and applicability permitting, one may be able to incorporate some extra topics to do as a class (or by group). For example, the class may decide to graph the theoretical and empirical pmf s for the given scenarios. This would give students a nice visual demonstration of how close the observed data is to the expected. In addition, if one would have designed an unfair deck, students would get the opportunity to notice the discrepancies between the theoretical and observed values. At the end of the activity, students are given a follow-up assignment for this activity which involves an activity write-up. These write-ups are completed for each activity that is performed in the class and are similar in nature. The group is asked to include purpose, design, and analysis/results with discussion. Students are asked to complete a write-up to give them experience on how to communicate statistical results in a meaningful way as well as to give them ownership of the results. Questions used for the write-up for this activity can be found at the end of Appendix A. As students completed the activity, some did not take it all that seriously (which happens during many in-class activities), while some experienced the light bulb moment. However, having the students work together has been anecdotally observed as beneficial as the students get the chance to explain, or question, a distribution to another student in the course. We believe that there is a difference in the level of understanding when students can explain a topic to other students versus merely being able to perform it themselves. In addition, students get confirmation when they reach the same conclusion as another student. 4. Assessment Design After implementing this activity a few times, it became of interest to know the extent to which this activity was helping students distinguish among these three distributions. In order to assess 6

8 Table 1: Timeline Day Class Material Assessments Given/Collected Wednesday Lecture on Binomial and Hypergeometric Distributions Monday Wednesday Monday Lecture on Negative Binomial; Practice Problems Activity Review for Exam Homework assignment given at the end of class Checked-off homework assignment Post-test given at the beginning of class Pre-test given at the beginning of class Activity Write-up due The option to leave a question blank complicated the scoring method as a blank answer (lack of knowledge) was believed to be different than an incorrect answer (wrong knowledge). The scoring method that was utilized was adapted from the SAT scoring method (College Board 2012): Correct answer: 1 point Incorrect answer: -0.5 point Blank answer: 0 point When utilizing this method, the highest score would be 8 points and the lowest score would be -4 points. 5. Results and Discussion At the time of this activity in the class, there were 25 students who were still registered for the course, 22 of whom gave permission for their data to be used and completed the pre-test. Two of these students were not in class the day of the post-test so their results are not included. Thus, the data are reported for 20 students with results for both the pre- and post-tests (see Table 2 below). 8

9 Table 2: Individual Student Results Version PRE-TEST POST-TEST for Pre- Test correct wrong blank Score correct wrong blank Score A A A A A A A A A A B B B B B B B B B B

10 Figure 1: Pre-Test/Post-Test Change Figure 1 is a graphical representation of how students scores changed from pre-test to post-test. A black arrow indicates a student whose score increased whereas a red arrow indicates a student whose score decreased. When looking at the Table 2 and Figure 1, one can see that there were quite a few students whose pre-test score is perfect (or near perfect). That is, there were five students who got at least seven of the questions correct on the pre-test. Thus seeing that some of their post-tests scores were lower is not as shocking since they didn t have as far to go up. However, there were also many students (11) who got at least 4 of the questions incorrect on the pre-test. Overall, the majority of the incorrect answers on both versions of the tests were Binomial leading authors to believe that this is the default answer to a student in doubt. We examined the scores to determine if post-test scores were significantly higher than pre-test scores. In order to assess this, the Wilcoxon Signed-Rank Sum Test was conducted and was found to be statistically significant (p-value = ). Although the ordering of the versions was randomized between students, it was still of interest to see if there was a significant difference in the change in scores from the pre-test to the post-test depending on which version the student was given as the pre-test. This was found to not be statistically significant (p-value = ) again using the Wilcoxon Signed-Rank Sum Test. As the semester progressed I discovered that my least successful students in the class all had version B first which can partly explain the observed difference in version ordering. 10

12 Appendix A: Activity Worksheet Activity Working with Discrete Distributions Scenario 1: With your 52-card deck you will shuffle, randomly draw a card, replace it, shuffle again, randomly draw a card, replace it, We will be interested in Y = the number of draws to obtain 2 hearts. 1. Repeat this process 10 times. (Thus, you should have 10 numbers that represent the number of draws it took to get 2 hearts). Record (tally) your values on the board with the rest of the class. 2. Based on the appropriate pmf, calculate the probability of getting the 2 nd heart on the 7 th draw. 3. Based upon our sample (on the board), what proportion of the time did the 2 nd heart come on the 7 th draw? Is this close to the answer in 2? Explain! 4. For this scenario, what is the expected number of draws to get the 2 nd heart? 5. For this scenario, what is the standard deviation for the number of draws to get the 2 nd heart? Use this as well as your answer in #4 above to interpret the standard deviation. Scenario 2: With your 52-card deck you will shuffle, randomly draw a card, replace it, shuffle again, randomly draw a card, replace it We will be interested in Y = the number of hearts drawn in 5 draws. 1. Repeat this process 10 times. (Thus, you should have 10 numbers that represent the number of hearts drawn in 5 draws). Record (tally) your values on the board with the rest of the class. 2. Based on the appropriate pmf, calculate the probability of getting exactly 1 heart in 5 draws. 3. Based upon our sample (on the board), what proportion of time did we get exactly 1 heart in 5 draws? Is this close to the answer in 2? Explain! 4. For this scenario, what is the expected number of hearts in 5 draws? 5. For this scenario, what is the standard deviation of the number of hearts in 5 draws? Use this as well as your answer in #4 above to interpret the standard deviation. 12

14 Appendix B: Question Pool Note: Questions 1 8 were Version A and questions 9-16 were Version B. Each question had the following options: Binomial Hypergeometric Negative Binomial 1. At a certain manufacturing company, approximately 5% of the products are defective. We are interested in calculating the probability that the third defective is the 20 th one sampled. 2. An assembly line produces products that they put into boxes of 50. The inspector then randomly picks 3 items inside a box to test to see if they are defective. In a box containing 4 defectives, they are interested in the probability that at least one of the three items sampled is defective. 3. When rolling a pair of dice, we generally care about the sum of the two dice. We are interested in the number of rolls that we perform before we get our first sum of The ACT is a standardized test that many students take in order to enter college. It is said that 4 out of every 5 students at NKU take the ACT. We are interested in the number of students in a given class of 30 that took the ACT. 5. A husband has 7 tasks on his to-do list and a wife has 10 things on her to-do list. Five tasks are randomly picked out of these 17 tasks. We are interest in the expected number of tasks the wife will have to do. 6. A certain stoplight, when coming from the North, is green approximately 31% of the time. Over the next few days, someone comes to this light 8 times from the North. We are interested in finding the expected number of green lights the person will come to. 7. A certain radio station s phone lines are all busy approximately 98% of the time when trying to call during a contest. We are interested in finding the probability that the 5 th time that you call is the 1 st time you get through during a contest. 8. Type O blood is one of the best to be donated since it can be used for many people. Approximately 42% of people have type O blood. In a given day at a blood bank, about 120 people come in to donate. The blood bank is interested in the number of donors who are type O. 9. A police officer has found that approximately 15% of the vehicles he pulls over are from out of state. We are interested in the number of vehicles that are out of state from the next 50 vehicles that he pulls over. 14

15 10. In a capture-recapture experiment, 20 animals were captured, tagged and released. A few weeks later, a sample of 40 of these animals is captured and we are interested in the number of animals in our sample that are tagged. 11. When taking the written drivers license test, they say that about 7 out of 8 people pass the test. A test-taker is interested in the number of times they will have to take the test in order to pass. 12. There is an urn containing 15 balls, 40% of which are green. A person gets to pull three balls out at the same time. For each ball that is green, he/she wins \$200. For each ball that is not green, he/she must pay \$50. We are interested in the number of green balls pulled out of the A student is taking a true/false test that consists of 15 questions. The student has approximately a 72% chance of getting any individual question correct. We are interested in the probability that the student gets at least 9 of the 15 questions correct. 14. A company has a pool of 15 applicants (10 male, 5 female) for a particular position that has 3 current openings. They are interested in the probability that none of the positions are filled by females. 15. When rolling a pair of dice, we generally care about the sum of the two dice. We are interested in how many times we get a sum of 7 out of 30 rolls. 16. A couple likes to play darts together. However, the female is not as good at the game and only has about a 8% chance of winning any individual game. Being stubborn, the couple will play until the female wins two games. We are interested in the expected number of games the couple will need to play. Acknowledgements This material is based upon work supported by the National Science Foundation s STEM Talent Expansion Program (DUE-STEP) program under Grant No Any opinions, findings and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation. References Aliaga, M., Cuff, C., Garfield, J., Lock, R., Utts, J. and Witmer, J. (2005), Guidelines for Assessment and Instruction in Statistics Education (GAISE): College Report. American Statistical Association. Available at: 15

16 Chance, B. L. (1997), Experiences with Authentic Assessment Techniques in an Introductory Statistics Course, Journal of Statistics Education, 5(3). Available at: College Board. (2012), How the SAT is Scored. Last accessed on March 11, 2012 from: delmas, R. C., Garfield, J., & Chance, B. (1999), A Model of Classroom Research in Action: Developing Simulation Activities to Improve Students Statistical Reasoning, Journal of Statistics Education, 7(3). Available at: Ledolter, J. (1995), Projects in Introductory Statistics Courses, The American Statistician, 49, Lunsford, M. L., Rowell, G. H., & Goodson-Espy, T. (2006), Classroom Research: Assessment of Student Understanding of Sampling Distributions of Means and the Central Limit Theorem in Post-Calculus Probability and Statistics Classes, Journal of Statistics Education, 14(3). Available at: Mills, J. D. (2002), Using Computer Simulation Methods to Teach Statistics: A Review of the Literature, Journal of Statistics Education, 10(1). Available at: Pfaff, T. J. & Weinberg, A. (2009), Do Hands-On Activities Increase Student Understanding?: A Case Study, Journal of Statistics Education, 17(3). Available at: Wackerly, D.D., Mendenhall, W., & Scheaffer, R.L. (2008), Mathematical Statistics with Applications, Belmont, CA: Thomson Brooks/Cole. Jacqueline Wroughton Nunn Drive MP 422 Highland Heights, KY Phone: (859) Tarah Cole 311 Upham Hall Oxford, Ohio Volume 22 (2013) Archive Index Data Archive Resources Editorial Board Guidelines for Authors Guidelines for Data Contributors Guidelines for Readers/Data Users Home Page Contact JSE ASA Publications 16

### Lecture 14. Chapter 7: Probability. Rule 1: Rule 2: Rule 3: Nancy Pfenning Stats 1000

Lecture 4 Nancy Pfenning Stats 000 Chapter 7: Probability Last time we established some basic definitions and rules of probability: Rule : P (A C ) = P (A). Rule 2: In general, the probability of one event

### Section 6.1 Discrete Random variables Probability Distribution

Section 6.1 Discrete Random variables Probability Distribution Definitions a) Random variable is a variable whose values are determined by chance. b) Discrete Probability distribution consists of the values

### Understanding Confidence Intervals and Hypothesis Testing Using Excel Data Table Simulation

Understanding Confidence Intervals and Hypothesis Testing Using Excel Data Table Simulation Leslie Chandrakantha lchandra@jjay.cuny.edu Department of Mathematics & Computer Science John Jay College of

### Truman College-Mathematics Department Math 125-CD: Introductory Statistics Course Syllabus Fall 2012

Instructor: Dr. Abdallah Shuaibi Office #: 3816 Email: ashuaibi1@ccc.edu URL: http://faculty.ccc.edu/ashuaibi/ Phone #: (773)907-4085 Office Hours: Truman College-Mathematics Department Math 125-CD: Introductory

### STAT 35A HW2 Solutions

STAT 35A HW2 Solutions http://www.stat.ucla.edu/~dinov/courses_students.dir/09/spring/stat35.dir 1. A computer consulting firm presently has bids out on three projects. Let A i = { awarded project i },

### Statistics 100A Homework 4 Solutions

Chapter 4 Statistics 00A Homework 4 Solutions Ryan Rosario 39. A ball is drawn from an urn containing 3 white and 3 black balls. After the ball is drawn, it is then replaced and another ball is drawn.

### The Math. P (x) = 5! = 1 2 3 4 5 = 120.

The Math Suppose there are n experiments, and the probability that someone gets the right answer on any given experiment is p. So in the first example above, n = 5 and p = 0.2. Let X be the number of correct

### 9. Sampling Distributions

9. Sampling Distributions Prerequisites none A. Introduction B. Sampling Distribution of the Mean C. Sampling Distribution of Difference Between Means D. Sampling Distribution of Pearson's r E. Sampling

### Curriculum Map Statistics and Probability Honors (348) Saugus High School Saugus Public Schools 2009-2010

Curriculum Map Statistics and Probability Honors (348) Saugus High School Saugus Public Schools 2009-2010 Week 1 Week 2 14.0 Students organize and describe distributions of data by using a number of different

### MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

Ch. 4 Discrete Probability Distributions 4.1 Probability Distributions 1 Decide if a Random Variable is Discrete or Continuous 1) State whether the variable is discrete or continuous. The number of cups

### AP Statistics 7!3! 6!

Lesson 6-4 Introduction to Binomial Distributions Factorials 3!= Definition: n! = n( n 1)( n 2)...(3)(2)(1), n 0 Note: 0! = 1 (by definition) Ex. #1 Evaluate: a) 5! b) 3!(4!) c) 7!3! 6! d) 22! 21! 20!

### Chapter 10: Introducing Probability

Chapter 10: Introducing Probability Randomness and Probability So far, in the first half of the course, we have learned how to examine and obtain data. Now we turn to another very important aspect of Statistics

### Lab 11. Simulations. The Concept

Lab 11 Simulations In this lab you ll learn how to create simulations to provide approximate answers to probability questions. We ll make use of a particular kind of structure, called a box model, that

### Mind on Statistics. Chapter 8

Mind on Statistics Chapter 8 Sections 8.1-8.2 Questions 1 to 4: For each situation, decide if the random variable described is a discrete random variable or a continuous random variable. 1. Random variable

### Overview. Essential Questions. Grade 7 Mathematics, Quarter 4, Unit 4.2 Probability of Compound Events. Number of instruction days: 8 10

Probability of Compound Events Number of instruction days: 8 10 Overview Content to Be Learned Find probabilities of compound events using organized lists, tables, tree diagrams, and simulation. Understand

### Performance Assessment Task Fair Game? Grade 7. Common Core State Standards Math - Content Standards

Performance Assessment Task Fair Game? Grade 7 This task challenges a student to use understanding of probabilities to represent the sample space for simple and compound events. A student must use information

### Chapter 4 Lecture Notes

Chapter 4 Lecture Notes Random Variables October 27, 2015 1 Section 4.1 Random Variables A random variable is typically a real-valued function defined on the sample space of some experiment. For instance,

### Chapter 3: DISCRETE RANDOM VARIABLES AND PROBABILITY DISTRIBUTIONS

Chapter 3: DISCRETE RANDOM VARIABLES AND PROBABILITY DISTRIBUTIONS Part 4: Geometric Distribution Negative Binomial Distribution Hypergeometric Distribution Sections 3-7, 3-8 The remaining discrete random

### MATHEMATICS FOR ENGINEERS STATISTICS TUTORIAL 4 PROBABILITY DISTRIBUTIONS

MATHEMATICS FOR ENGINEERS STATISTICS TUTORIAL 4 PROBABILITY DISTRIBUTIONS CONTENTS Sample Space Accumulative Probability Probability Distributions Binomial Distribution Normal Distribution Poisson Distribution

### Math 58. Rumbos Fall 2008 1. Solutions to Review Problems for Exam 2

Math 58. Rumbos Fall 2008 1 Solutions to Review Problems for Exam 2 1. For each of the following scenarios, determine whether the binomial distribution is the appropriate distribution for the random variable

### The Casino Lab STATION 1: CRAPS

The Casino Lab Casinos rely on the laws of probability and expected values of random variables to guarantee them profits on a daily basis. Some individuals will walk away very wealthy, while others will

### STAT 2300: BUSINESS STATISTICS Section 002, Summer Semester 2009

STAT 2300: BUSINESS STATISTICS Section 002, Summer Semester 2009 Instructor: Bill Welbourn Office: Lund 117 Email: bill.welbourn@aggiemail.usu.edu Lectures: MWF 7:30AM 9:40AM in ENGR 104 Office Hours:

### STAT 360 Probability and Statistics. Fall 2012

STAT 360 Probability and Statistics Fall 2012 1) General information: Crosslisted course offered as STAT 360, MATH 360 Semester: Fall 2012, Aug 20--Dec 07 Course name: Probability and Statistics Number

### UNIVERSITY of MASSACHUSETTS DARTMOUTH Charlton College of Business Decision and Information Sciences Fall 2010

UNIVERSITY of MASSACHUSETTS DARTMOUTH Charlton College of Business Decision and Information Sciences Fall 2010 COURSE: POM 500 Statistical Analysis, ONLINE EDITION, Fall 2010 Prerequisite: Finite Math

### A Hands-On Exercise Improves Understanding of the Standard Error. of the Mean. Robert S. Ryan. Kutztown University

A Hands-On Exercise 1 Running head: UNDERSTANDING THE STANDARD ERROR A Hands-On Exercise Improves Understanding of the Standard Error of the Mean Robert S. Ryan Kutztown University A Hands-On Exercise

### Session 8 Probability

Key Terms for This Session Session 8 Probability Previously Introduced frequency New in This Session binomial experiment binomial probability model experimental probability mathematical probability outcome

### 13 Two-Sample T Tests

www.ck12.org CHAPTER 13 Two-Sample T Tests Chapter Outline 13.1 TESTING A HYPOTHESIS FOR DEPENDENT AND INDEPENDENT SAMPLES 270 www.ck12.org Chapter 13. Two-Sample T Tests 13.1 Testing a Hypothesis for

### On October 14, 2010, I taught a social studies lesson about Latitude and Longitude to 5 th

Phylicia Kelly Professor Schilling EDUC. 327 October 21, 2010 Student Teaching Reflection On October 14, 2010, I taught a social studies lesson about Latitude and Longitude to 5 th grade students at my

### Hey, That s Not Fair! (Or is it?)

Concept Probability and statistics Number sense Activity 9 Hey, That s Not Fair! (Or is it?) Students will use the calculator to simulate dice rolls to play two different games. They will decide if the

### Polynomials and Factoring. Unit Lesson Plan

Polynomials and Factoring Unit Lesson Plan By: David Harris University of North Carolina Chapel Hill Math 410 Dr. Thomas, M D. 2 Abstract This paper will discuss, and give, lesson plans for all the topics

### Gaming the Law of Large Numbers

Gaming the Law of Large Numbers Thomas Hoffman and Bart Snapp July 3, 2012 Many of us view mathematics as a rich and wonderfully elaborate game. In turn, games can be used to illustrate mathematical ideas.

### 6.042/18.062J Mathematics for Computer Science. Expected Value I

6.42/8.62J Mathematics for Computer Science Srini Devadas and Eric Lehman May 3, 25 Lecture otes Expected Value I The expectation or expected value of a random variable is a single number that tells you

### Permutations and Combinations (Assisted with TI-83+ silver)

Dan Tracz Grade 10/11 5 day lesson plan 11/7/04 Permutations and Combinations (Assisted with TI-83+ silver) Objectives for this unit - To create a better understanding of combinations and permutations.

### 7. Normal Distributions

7. Normal Distributions A. Introduction B. History C. Areas of Normal Distributions D. Standard Normal E. Exercises Most of the statistical analyses presented in this book are based on the bell-shaped

### CONTENTS OF DAY 2. II. Why Random Sampling is Important 9 A myth, an urban legend, and the real reason NOTES FOR SUMMER STATISTICS INSTITUTE COURSE

1 2 CONTENTS OF DAY 2 I. More Precise Definition of Simple Random Sample 3 Connection with independent random variables 3 Problems with small populations 8 II. Why Random Sampling is Important 9 A myth,

### Chapter 4. Probability Distributions

Chapter 4 Probability Distributions Lesson 4-1/4-2 Random Variable Probability Distributions This chapter will deal the construction of probability distribution. By combining the methods of descriptive

### Title: Transforming a traditional lecture-based course to online and hybrid models of learning

Title: Transforming a traditional lecture-based course to online and hybrid models of learning Author: Susan Marshall, Lecturer, Psychology Department, Dole Human Development Center, University of Kansas.

### MA 1125 Lecture 14 - Expected Values. Friday, February 28, 2014. Objectives: Introduce expected values.

MA 5 Lecture 4 - Expected Values Friday, February 2, 24. Objectives: Introduce expected values.. Means, Variances, and Standard Deviations of Probability Distributions Two classes ago, we computed the

### Final Exam Performance. 50 OLI Accel Trad Control Trad All. Figure 1. Final exam performance of accelerated OLI-Statistics compared to traditional

IN SEARCH OF THE PERFECT BLEND BETWEEN AN INSTRUCTOR AND AN ONLINE COURSE FOR TEACHING INTRODUCTORY STATISTICS Marsha Lovett, Oded Meyer and Candace Thille Carnegie Mellon University, United States of

### Preparation of Two-Year College Mathematics Instructors to Teach Statistics with GAISE Session on Assessment

Preparation of Two-Year College Mathematics Instructors to Teach Statistics with GAISE Session on Assessment - 1 - American Association for Higher Education (AAHE) 9 Principles of Good Practice for Assessing

### Elementary Statistics and Inference. Elementary Statistics and Inference. 16 The Law of Averages (cont.) 22S:025 or 7P:025.

Elementary Statistics and Inference 22S:025 or 7P:025 Lecture 20 1 Elementary Statistics and Inference 22S:025 or 7P:025 Chapter 16 (cont.) 2 D. Making a Box Model Key Questions regarding box What numbers

### AMS 5 CHANCE VARIABILITY

AMS 5 CHANCE VARIABILITY The Law of Averages When tossing a fair coin the chances of tails and heads are the same: 50% and 50%. So if the coin is tossed a large number of times, the number of heads and

### 4. Joint Distributions

Virtual Laboratories > 2. Distributions > 1 2 3 4 5 6 7 8 4. Joint Distributions Basic Theory As usual, we start with a random experiment with probability measure P on an underlying sample space. Suppose

### Exam 3 Review/WIR 9 These problems will be started in class on April 7 and continued on April 8 at the WIR.

Exam 3 Review/WIR 9 These problems will be started in class on April 7 and continued on April 8 at the WIR. 1. Urn A contains 6 white marbles and 4 red marbles. Urn B contains 3 red marbles and two white

### 6.1. Construct and Interpret Binomial Distributions. p Study probability distributions. Goal VOCABULARY. Your Notes.

6.1 Georgia Performance Standard(s) MM3D1 Your Notes Construct and Interpret Binomial Distributions Goal p Study probability distributions. VOCABULARY Random variable Discrete random variable Continuous

Chapter 11 Testing Hypotheses About Proportions Hypothesis testing method: uses data from a sample to judge whether or not a statement about a population may be true. Steps in Any Hypothesis Test 1. Determine

### Flipped Statistics Class Results: Better Performance Than Lecture Over One Year Later

Flipped Statistics Class Results: Better Performance Than Lecture Over One Year Later Jennifer R. Winquist Kieth A. Carlson Valparaiso University Journal of Statistics Education Volume 22, Number 3 (2014),

### Chapter 5. Section 5.1: Central Tendency. Mode: the number or numbers that occur most often. Median: the number at the midpoint of a ranked data.

Chapter 5 Section 5.1: Central Tendency Mode: the number or numbers that occur most often. Median: the number at the midpoint of a ranked data. Example 1: The test scores for a test were: 78, 81, 82, 76,

### WHERE DOES THE 10% CONDITION COME FROM?

1 WHERE DOES THE 10% CONDITION COME FROM? The text has mentioned The 10% Condition (at least) twice so far: p. 407 Bernoulli trials must be independent. If that assumption is violated, it is still okay

### Fundamentals of Probability

Fundamentals of Probability Introduction Probability is the likelihood that an event will occur under a set of given conditions. The probability of an event occurring has a value between 0 and 1. An impossible

### Statistics 100A Homework 3 Solutions

Chapter Statistics 00A Homework Solutions Ryan Rosario. Two balls are chosen randomly from an urn containing 8 white, black, and orange balls. Suppose that we win \$ for each black ball selected and we

### ACTM State Exam-Statistics

ACTM State Exam-Statistics For the 25 multiple-choice questions, make your answer choice and record it on the answer sheet provided. Once you have completed that section of the test, proceed to the tie-breaker

### Outcomes of Preservice Teacher s Technology Use

Outcomes of Preservice Teacher s Technology Use William F. Morrison, Assistant Professor, Bowling Green State University Tara L. Jeffs, Assistant Professor, East Carolina University Abstract: At a time

### Data Analysis, Statistics, and Probability

Chapter 6 Data Analysis, Statistics, and Probability Content Strand Description Questions in this content strand assessed students skills in collecting, organizing, reading, representing, and interpreting

### Onsite Peer Tutoring in Mathematics Content Courses for Pre-Service Teachers

IUMPST: The Journal. Vol 2 (Pedagogy), February 2011. [www.k-12prep.math.ttu.edu] Onsite Peer Tutoring in Mathematics Content Courses for Pre-Service Teachers Elaine Young Associate Professor of Mathematics

### Statistics 100A Homework 2 Solutions

Statistics Homework Solutions Ryan Rosario Chapter 9. retail establishment accepts either the merican Express or the VIS credit card. total of percent of its customers carry an merican Express card, 6

### Introduction to the Practice of Statistics Fifth Edition Moore, McCabe

Introduction to the Practice of Statistics Fifth Edition Moore, McCabe Section 5.1 Homework Answers 5.7 In the proofreading setting if Exercise 5.3, what is the smallest number of misses m with P(X m)

### Chapter 4 & 5 practice set. The actual exam is not multiple choice nor does it contain like questions.

Chapter 4 & 5 practice set. The actual exam is not multiple choice nor does it contain like questions. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

### Grade 7 Equivalent Ratios Lesson and Resources

Grade 7 Equivalent Ratios Lesson and Resources VSC Standard 6.0 Knowledge of Number Relationships and Computations/Arithmetic Students will describe, represent, or apply numbers or their relationships

### The Binomial Probability Distribution

The Binomial Probability Distribution MATH 130, Elements of Statistics I J. Robert Buchanan Department of Mathematics Fall 2015 Objectives After this lesson we will be able to: determine whether a probability

### MATH 140 HYBRID INTRODUCTORY STATISTICS COURSE SYLLABUS

MATH 140 HYBRID INTRODUCTORY STATISTICS COURSE SYLLABUS Instructor: Mark Schilling Email: mark.schilling@csun.edu (Note: If your CSUN email address is not one you use regularly, be sure to set up automatic

### The mathematical branch of probability has its

ACTIVITIES for students Matthew A. Carlton and Mary V. Mortlock Teaching Probability and Statistics through Game Shows The mathematical branch of probability has its origins in games and gambling. And

### 1) The table lists the smoking habits of a group of college students. Answer: 0.218

FINAL EXAM REVIEW Name ) The table lists the smoking habits of a group of college students. Sex Non-smoker Regular Smoker Heavy Smoker Total Man 5 52 5 92 Woman 8 2 2 220 Total 22 2 If a student is chosen

### Introduction and Overview

Introduction and Overview Probability and Statistics is a topic that is quickly growing, has become a major part of our educational program, and has a substantial role in the NCTM Standards. While covering

### Worksheet for Teaching Module Probability (Lesson 1)

Worksheet for Teaching Module Probability (Lesson 1) Topic: Basic Concepts and Definitions Equipment needed for each student 1 computer with internet connection Introduction In the regular lectures in

### Probability and Statistics

Probability and Statistics Activity: TEKS: (8.11) Probability and statistics. The student applies concepts of theoretical and experimental probability to make predictions. The student is expected to: (C)

### STUDENTS PROBABILSTIC SIMULATION AND MODELING COMPETENCE AFTER A COMPUTER-INTENSIVE ELEMENTARY COURSE IN STATISTICS AND PROBABILITY

STUDENTS PROBABILSTIC SIMULATION AND MODELING COMPETENCE AFTER A COMPUTER-INTENSIVE ELEMENTARY COURSE IN STATISTICS AND PROBABILITY Carmen Maxara and Rolf Biehler University of Kassel, Germany carmen@mathematik.uni-kassel.de

### Levels of measurement in psychological research:

Research Skills: Levels of Measurement. Graham Hole, February 2011 Page 1 Levels of measurement in psychological research: Psychology is a science. As such it generally involves objective measurement of

### Chapter 4: Probabilities and Proportions

Stats 11 (Fall 2004) Lecture Note Introduction to Statistical Methods for Business and Economics Instructor: Hongquan Xu Chapter 4: Probabilities and Proportions Section 4.1 Introduction In the real world,

### Spring 2013 Structured Learning Assistance (SLA) Program Evaluation Results

Crafton Hills College RRN 682 July 2013 Research Brief Spring 2013 Structured Learning Assistance (SLA) Program Evaluation Results Prepared by Lorena Guadiana Summary of Main Findings 85% of respondents

### PRACTICE BOOK MATHEMATICS TEST (RESCALED) Graduate Record Examinations. This practice book contains. Become familiar with

This book is provided FREE with test registration by the Graduate Record Examinations Board. Graduate Record Examinations This practice book contains one actual full-length GRE Mathematics Test (Rescaled)

### Probability Theory, Part 4: Estimating Probabilities from Finite Universes

8 Resampling: The New Statistics CHAPTER 8 Probability Theory, Part 4: Estimating Probabilities from Finite Universes Introduction Some Building Block Programs Problems in Finite Universes Summary Introduction

### Unit 7 Quadratic Relations of the Form y = ax 2 + bx + c

Unit 7 Quadratic Relations of the Form y = ax 2 + bx + c Lesson Outline BIG PICTURE Students will: manipulate algebraic expressions, as needed to understand quadratic relations; identify characteristics

### PROBABILITY SECOND EDITION

PROBABILITY SECOND EDITION Table of Contents How to Use This Series........................................... v Foreword..................................................... vi Basics 1. Probability All

### Results from the 2014 AP Statistics Exam. Jessica Utts, University of California, Irvine Chief Reader, AP Statistics jutts@uci.edu

Results from the 2014 AP Statistics Exam Jessica Utts, University of California, Irvine Chief Reader, AP Statistics jutts@uci.edu The six free-response questions Question #1: Extracurricular activities

### Data Coding and Entry Lessons Learned

Chapter 7 Data Coding and Entry Lessons Learned Pércsich Richárd Introduction In this chapter we give an overview of the process of coding and entry of the 1999 pilot test data for the English examination

### Random variables, probability distributions, binomial random variable

Week 4 lecture notes. WEEK 4 page 1 Random variables, probability distributions, binomial random variable Eample 1 : Consider the eperiment of flipping a fair coin three times. The number of tails that

### Lesson Plan #3. Simulations

Lesson Plan #3 Simulations Grade: 7 th grade Subject: Mathematics About the class: Standards: Objectives: - 20 students- 11 boys, 9 girls - 7 students are special education students - One-teach, one-assist

### Thursday, November 13: 6.1 Discrete Random Variables

Thursday, November 13: 6.1 Discrete Random Variables Read 347 350 What is a random variable? Give some examples. What is a probability distribution? What is a discrete random variable? Give some examples.

### The Loss of Intuition - A Lesson for the School Teacher?

The Loss of Intuition - A Lesson for the School Teacher? Flavia R Jolliffe - Surrey, England 1. The study Although there is a long tradition of research into concepts and intuitions regarding randomness

### Name: Date: Use the following to answer questions 3-4:

Name: Date: 1. Determine whether each of the following statements is true or false. A) The margin of error for a 95% confidence interval for the mean increases as the sample size increases. B) The margin

### Summary of Probability

Summary of Probability Mathematical Physics I Rules of Probability The probability of an event is called P(A), which is a positive number less than or equal to 1. The total probability for all possible

### Question 1 Formatted: Formatted: Formatted: Formatted:

In many situations in life, we are presented with opportunities to evaluate probabilities of events occurring and make judgments and decisions from this information. In this paper, we will explore four

### The practice test follows this cover sheet. It is very similar to the real Chapter Test.

AP Stats Unit IV (Chapters 14-17) Take-Home Test Info The practice test follows this cover sheet. It is very similar to the real Chapter 14-17 Test. The real test will consist of 20 multiple-choice questions

### Chapter 6 Continuous Probability Distributions

Continuous Probability Distributions Learning Objectives 1. Understand the difference between how probabilities are computed for discrete and continuous random variables. 2. Know how to compute probability

### Data Analysis with Surveys

Grade Level: 3 Time: Three 45 min class periods By: Keith Barton (Bedford County Public Schools) In this lesson, the students will create and conduct a survey. They will then collect and analyze the data

### COS 160: Course Assessment

COS 160: Course Assessment Student Responses from Focus Group Sessions Spring 2005 Office of Academic Assessment University of Southern Maine Spring 2005 Introduction The Computer Science department was

### Internet classes are being seen more and more as

Internet Approach versus Lecture and Lab-Based Approach Blackwell Oxford, TEST Teaching 0141-982X Journal Original XXXXXXXXX 2008 The compilation UK Articles Statistics Publishing Authors Ltd 2008 Teaching

### Cell Phone Impairment?

Cell Phone Impairment? Overview of Lesson This lesson is based upon data collected by researchers at the University of Utah (Strayer and Johnston, 2001). The researchers asked student volunteers (subjects)

### Problem sets for BUEC 333 Part 1: Probability and Statistics

Problem sets for BUEC 333 Part 1: Probability and Statistics I will indicate the relevant exercises for each week at the end of the Wednesday lecture. Numbered exercises are back-of-chapter exercises from

### Steve Sworder Mathematics Department Saddleback College Mission Viejo, CA 92692 ssworder@saddleback.edu. August, 2007

Comparison of the Effectiveness of a Traditional Intermediate Algebra Course With That of a Less Rigorous Intermediate Algebra Course in Preparing Students for Success in a Subsequent Mathematics Course

### Chapter 3: DISCRETE RANDOM VARIABLES AND PROBABILITY DISTRIBUTIONS. Part 3: Discrete Uniform Distribution Binomial Distribution

Chapter 3: DISCRETE RANDOM VARIABLES AND PROBABILITY DISTRIBUTIONS Part 3: Discrete Uniform Distribution Binomial Distribution Sections 3-5, 3-6 Special discrete random variable distributions we will cover

### Statistics 104: Section 6!

Page 1 Statistics 104: Section 6! TF: Deirdre (say: Dear-dra) Bloome Email: dbloome@fas.harvard.edu Section Times Thursday 2pm-3pm in SC 109, Thursday 5pm-6pm in SC 705 Office Hours: Thursday 6pm-7pm SC

### LIVE CASE STUDIES IN ORGANIZATIONAL CHANGE: LEARNING ABOUT CHANGE THROUGH STUDENT PHILANTHROPY AND SERVICE-LEARNING

International Journal of Case Method Research & Application (2006) XVIII, 2 2006 WACRA. All rights reserved ISSN 1554-7752 LIVE CASE STUDIES IN ORGANIZATIONAL CHANGE: LEARNING ABOUT CHANGE THROUGH STUDENT

### 13.0 Central Limit Theorem

13.0 Central Limit Theorem Discuss Midterm/Answer Questions Box Models Expected Value and Standard Error Central Limit Theorem 1 13.1 Box Models A Box Model describes a process in terms of making repeated

### Introductory Probability. MATH 107: Finite Mathematics University of Louisville. March 5, 2014

Introductory Probability MATH 07: Finite Mathematics University of Louisville March 5, 204 What is probability? Counting and probability 2 / 3 Probability in our daily lives We see chances, odds, and probabilities

### Investigating the Investigative Task: Testing for Skewness An Investigation of Different Test Statistics and their Power to Detect Skewness

Investigating the Investigative Task: Testing for Skewness An Investigation of Different Test Statistics and their Power to Detect Skewness Josh Tabor Canyon del Oro High School Journal of Statistics Education