# MONT 107N Understanding Randomness Solutions For Final Examination May 11, 2010

Size: px
Start display at page:

Download "MONT 107N Understanding Randomness Solutions For Final Examination May 11, 2010"

Transcription

1 MONT 07N Understanding Randomness Solutions For Final Examination May, 00 Short Answer (a) (0) How are the EV and SE for the sum of n draws with replacement from a box computed? Solution: The EV is n times the average of the tickets in the box The SE is n times the SD of the tickets in the box (b) (0) What does the Central Limit Theorem say? Solution: The Central Limit Theorem says that the probability histogram for sum of n draws with replacement from any box will tend to a (shifted, scaled) normal curve as n, in the sense that the area in the histogram for for any range of sum values will tend to the area under the corresponding normal curve (c) (0) What is a 99% confidence interval for an average? Solution: The 99% confidence interval is the interval sample average ± 3 (d) (0) What is the p-value of a test of significance? sample SD n Solution: The p-value is the chance that the observed value of the test statistic, or something more extreme, would occur if the null hypothesis is true (e) (0) When is a t-test for an average used instead of a z-test? The t-test is used when the size of the sample is small (usually n < 30), and it is known that the sampled values come from a population that has a normal distribution A statistician tosses a coin 00 times His null hypothesis is that the coin is fair, and his alternative is that the coin is biased in favor of heads The coin comes up heads 60 times For each of the following, say whether the statement is true or false, and explain (a) (0) If the coin was fair, there would be about a 3% chance of getting 60 or more heads in 00 tosses Solution: This is TRUE Making 00 tosses of a fair coin is like drawing 00 times with replacement from the box [0,] The expected number of heads is 00 = 50, and the SE for the number of heads is 00 = 5 Using the normal approximation, the chance of getting 60 or more heads is approximated by the area in the upper tail of the normal curve starting at = 9 From the normal table, his area is (00% A(9)) = (00% 946%) = 87%, which is about 3% (Recall that in computing chances for numbers of discrete events using the normal curve, the corresponding area for a single number extends from that number minus 5 to that number plus 5 This is the continuity correction )

2 (b) (0) Given that 60 heads are observed, there is only about a 3% chance that the coin is fair Solution: This is FALSE A particular coin is either fair or not There is no chance there The chance is in the tossing, not in the coin itself 3 A gambler is playing roulette On each play, he bets \$ on a split (two adjacent numbers), which pays 7 to That is, if one of his numbers comes up, he will win \$7 on a \$ bet Otherwise, he loses the \$ (a) (5) Construct a box model for the payoff on this bet (Hint: Recall there are a total of 38 slots on a roulette wheel) Solution: The box is [7, 7, 36 ] (b) (0) Compute the average and SD of the box Solution: The average is ( ( ))/38 = 053 The SD, by the shortcut method, is: SD = (7 ( )) = (c) (0) If the bet is played 00 times, compute the EV and the SE for the payoff Solution: The EV is = 53 (that is the gambler is expected to lose about \$530 overall The SE is 00 \$40 = \$40 (d) (0) What are the gambler s chances of coming out ahead on 00 spins of the wheel? Solution: We want to estimate the chance that the net payoff is at least 0 In standard units, this corresponds to z = 0 ( 53) = 3 40 From the normal table, we can estimate the chance as being between (00% A(0)) = (00% 797) = 46% and (00% A(5)) = (00% 9) = 44% (Not bad, but the casino always wins in the end!) 4 An airline does a market research survey on travel patterns It takes a simple random sample of 5 people and works out the 95% confidence interval for the average distance they traveled on vacation in the previous year The computed interval extends from 488 to 59 miles Say whether each statement below is true or false and give reasons If there is not enough information given to decide, explain what other information you would need (a) (5) The average distance traveled was about 540 miles

3 Solution: TRUE As in question part c, the midpoint of the confidence interval is the observed average The midpoint is ( )/ = 540 (b) (5) The SD of the distances traveled was about 00 miles Solution: FALSE The confidence interval extends SD 5 in both directions from the midpoint 5 = SD 5 implies the SD had to be about 390 (c) (5) The probability histogram for averages of samples of size 5 drawn from the population is probably close to (scaled, shifted) normal curve Solution: TRUE This is far enough into the large sample range that the Central Limit Theorem would apply in almost every case (d) (5) The probability histogram for the distances traveled by randomly chosen individuals in the population is close to a (scaled, shifted) normal curve Solution: NEED MORE INFORMATION There is no way to tell whether that is true from the information given (Like many real-world distributions, this one could have a long right tail if there were a few people who traveled very long distances, while the majority traveled much smaller distances) (e) (5) If the sample size is increased to 450, the width of the confidence interval will always be about half as wide Solution: FALSE If the SD stayed the same, we would expect the width to decrease by a factor of = 707, not = 5 The SD of the sample could also change 5 A simple random sample of 000 persons is taken to determine the percentage of Democrats in a large population It turns out that 543 of the people in the sample are Democrats (a) (0) Compute a 95% confidence interval for the percentage of Democrats in the population Solution: The observed percentage in the sample is 543 by the bootstrap method: 000 = 543% The SE is estimated SE for percent = Then the confidence interval is (543)(457) % = 58% observed ± SE = 543% ± 3% (b) (0) True or False and explain: There is about a 5% chance that the actual percentage of Democrats is outside the interval found in part A Solution: FALSE The chance is in the sampling process, not in the location of the actual percentage of Democrats The correct interpretation is that about 5% of samples would yield intervals that did not contain the actual percentage of Democrats 3

4 6 The National Assessment of Educational Progress tested 7-year-olds on mathematics in 990 and again in 004 In 990, the average score was 305 and the SD was 34 In 004, the average score was 307 and the SD was 7 In both years, simple random samples of size 000 were used Can the difference be explained as chance variation, or have average mathematics scores really gone up? (a) (5) Should you use a one-sample or a two-sample z-test for this kind of question? Solution: This calls for a two-sample z-test, since there are two completely independent groups of students who would have taken the tests in the two different years (b) (0) Formulate null and alternative hypotheses in terms of a box model (Say how many boxes there are, how many tickets go in each, and what is on the tickets scores or 0/ s) Solution: There are two boxes, one for the population of students that took the test in each year There is a ticket for each student each time The tickets have their scores, not 0/ The null hypothesis would be that the averages of the two boxes are the same; the alternative is that the 004 average is higher (c) (0) Carry out the test, find the associated p-value, and discuss your results Solution: We use the -sample z-test formulas The test statistic is z = (35) (7) 000 = 43 From the normal table, we see that the p-value is found between (00% A(40)) = (00% 8385) = 8% p = 08 and (00% A(45)) = (00% 859) = 74% p = 074 Usually p-values like these are not considered small enough to reject the null hypothesis We conclude the difference could have been due to chance (And the -point difference in average score hardly seems practically significant in any case!) 7 A New York Times/CBS poll conducted between April 5 and April, 00 included the question Do you approve or disapprove of the way your Representative in Congress is handling his or her job? The poll was carried out in two stages First a large national simple random sample was asked the question The results were that 46% of respondents said they approved, 36% said they disapproved, and 8% said they had no opinion Then the same question was asked of a separate sample of 88 people, all of whom had identified themselves as supporters of the Tea Party movement The responses broke down like this: Approve Disapprove No opinion Total Tea Party Sample

5 (Data is adapted from the New York Times web site) It seems from the numbers that Tea Party supporters may differ from the general population when it comes to their opinions concerning their Congressional Representatives But is this a real difference, or could it just be a product of chance variation in the sampling process? (a) (5) Formulate a null and alternative hypothesis for this question Solution: The null hypothesis would be that there is really no difference, and the observed differences were due to chance The alternative would be that Tea Partiers really do have a different opinion about the jobs that their Congressional Representatives are doing (b) (5) What significance test should you use to decide between these? Solution: This calls for a χ test The idea is to see whether the data from the Tea Partiers matches the national percentages out of n = 88 In other words, do a goodness of fit test (c) (0) Carry out your test of signficance What do you get for a test statistic and observed significance level (p-value)? Solution: If the Tea Partiers were really the same as the general population in this regard, then the expected numbers of people in the Approve, Disapprove, and No opinion categories would be: = 4053, = 37, 8 88 = 586 We use these to compute the χ statistic using the entries from the second row: χ = ( ) (43 37) 37 + (97 586) 586 = 75 From the χ table with degrees of freedom, we see that p < 0 (much, much smaller!) (d) (5) Does this data give evidence of a real difference between Tea Party supporters and the public as a whole? Explain Solution: This is extremely strong evidence that Tea Partiers have a different set of opinions on this question (They are much less likely to approve, and also much less likely not to have an opinion!) Have a safe, enjoyable, and productive summer! 5

### 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

### Chapter 26: Tests of Significance

Chapter 26: Tests of Significance Procedure: 1. State the null and alternative in words and in terms of a box model. 2. Find the test statistic: z = observed EV. SE 3. Calculate the P-value: The area under

### Elementary Statistics and Inference. Elementary Statistics and Inference. 17 Expected Value and Standard Error. 22S:025 or 7P:025.

Elementary Statistics and Inference S:05 or 7P:05 Lecture Elementary Statistics and Inference S:05 or 7P:05 Chapter 7 A. The Expected Value In a chance process (probability experiment) the outcomes of

### The Normal Approximation to Probability Histograms. Dice: Throw a single die twice. The Probability Histogram: Area = Probability. Where are we going?

The Normal Approximation to Probability Histograms Where are we going? Probability histograms The normal approximation to binomial histograms The normal approximation to probability histograms of sums

### Stat 20: Intro to Probability and Statistics

Stat 20: Intro to Probability and Statistics Lecture 16: More Box Models Tessa L. Childers-Day UC Berkeley 22 July 2014 By the end of this lecture... You will be able to: Determine what we expect the sum

### t Tests in Excel The Excel Statistical Master By Mark Harmon Copyright 2011 Mark Harmon

t-tests in Excel By Mark Harmon Copyright 2011 Mark Harmon No part of this publication may be reproduced or distributed without the express permission of the author. mark@excelmasterseries.com www.excelmasterseries.com

### The overall size of these chance errors is measured by their RMS HALF THE NUMBER OF TOSSES NUMBER OF HEADS MINUS 0 400 800 1200 1600 NUMBER OF TOSSES

INTRODUCTION TO CHANCE VARIABILITY WHAT DOES THE LAW OF AVERAGES SAY? 4 coins were tossed 1600 times each, and the chance error number of heads half the number of tosses was plotted against the number

### John Kerrich s coin-tossing Experiment. Law of Averages - pg. 294 Moore s Text

Law of Averages - pg. 294 Moore s Text 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 the

### Class 19: Two Way Tables, Conditional Distributions, Chi-Square (Text: Sections 2.5; 9.1)

Spring 204 Class 9: Two Way Tables, Conditional Distributions, Chi-Square (Text: Sections 2.5; 9.) Big Picture: More than Two Samples In Chapter 7: We looked at quantitative variables and compared the

### 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

### 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

### 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

### Chapter 20: chance error in sampling

Chapter 20: chance error in sampling Context 2 Overview................................................................ 3 Population and parameter..................................................... 4

### Study Guide for the Final Exam

Study Guide for the Final Exam When studying, remember that the computational portion of the exam will only involve new material (covered after the second midterm), that material from Exam 1 will make

### \$2 4 40 + ( \$1) = 40

THE EXPECTED VALUE FOR THE SUM OF THE DRAWS In the game of Keno there are 80 balls, numbered 1 through 80. On each play, the casino chooses 20 balls at random without replacement. Suppose you bet on the

### Unit 27: Comparing Two Means

Unit 27: Comparing Two Means Prerequisites Students should have experience with one-sample t-procedures before they begin this unit. That material is covered in Unit 26, Small Sample Inference for One

### STA 130 (Winter 2016): An Introduction to Statistical Reasoning and Data Science

STA 130 (Winter 2016): An Introduction to Statistical Reasoning and Data Science Mondays 2:10 4:00 (GB 220) and Wednesdays 2:10 4:00 (various) Jeffrey Rosenthal Professor of Statistics, University of Toronto

### c. Construct a boxplot for the data. Write a one sentence interpretation of your graph.

MBA/MIB 5315 Sample Test Problems Page 1 of 1 1. An English survey of 3000 medical records showed that smokers are more inclined to get depressed than non-smokers. Does this imply that smoking causes depression?

### Chapter 16: law of averages

Chapter 16: law of averages Context................................................................... 2 Law of averages 3 Coin tossing experiment......................................................

### NCSS Statistical Software

Chapter 06 Introduction This procedure provides several reports for the comparison of two distributions, including confidence intervals for the difference in means, two-sample t-tests, the z-test, the

### Statistics and Random Variables. Math 425 Introduction to Probability Lecture 14. Finite valued Random Variables. Expectation defined

Expectation Statistics and Random Variables Math 425 Introduction to Probability Lecture 4 Kenneth Harris kaharri@umich.edu Department of Mathematics University of Michigan February 9, 2009 When a large

### Chapter 16. Law of averages. Chance. Example 1: rolling two dice Sum of draws. Setting up a. Example 2: American roulette. Summary.

Overview Box Part V Variability The Averages Box We will look at various chance : Tossing coins, rolling, playing Sampling voters We will use something called s to analyze these. Box s help to translate

### In the situations that we will encounter, we may generally calculate the probability of an event

What does it mean for something to be random? An event is called random if the process which produces the outcome is sufficiently complicated that we are unable to predict the precise result and are instead

### Solutions: Problems for Chapter 3. Solutions: Problems for Chapter 3

Problem A: You are dealt five cards from a standard deck. Are you more likely to be dealt two pairs or three of a kind? experiment: choose 5 cards at random from a standard deck Ω = {5-combinations of

### An Introduction to Statistics Course (ECOE 1302) Spring Semester 2011 Chapter 10- TWO-SAMPLE TESTS

The Islamic University of Gaza Faculty of Commerce Department of Economics and Political Sciences An Introduction to Statistics Course (ECOE 130) Spring Semester 011 Chapter 10- TWO-SAMPLE TESTS Practice

### Stat 20: Intro to Probability and Statistics

Stat 20: Intro to Probability and Statistics Lecture 18: Simple Random Sampling Tessa L. Childers-Day UC Berkeley 24 July 2014 By the end of this lecture... You will be able to: Draw box models for real-world

### Section 7C: The Law of Large Numbers

Section 7C: The Law of Large Numbers Example. You flip a coin 00 times. Suppose the coin is fair. How many times would you expect to get heads? tails? One would expect a fair coin to come up heads half

### Introduction to. Hypothesis Testing CHAPTER LEARNING OBJECTIVES. 1 Identify the four steps of hypothesis testing.

Introduction to Hypothesis Testing CHAPTER 8 LEARNING OBJECTIVES After reading this chapter, you should be able to: 1 Identify the four steps of hypothesis testing. 2 Define null hypothesis, alternative

### 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

### 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

### Two-sample inference: Continuous data

Two-sample inference: Continuous data Patrick Breheny April 5 Patrick Breheny STA 580: Biostatistics I 1/32 Introduction Our next two lectures will deal with two-sample inference for continuous data As

### Simulation Exercises to Reinforce the Foundations of Statistical Thinking in Online Classes

Simulation Exercises to Reinforce the Foundations of Statistical Thinking in Online Classes Simcha Pollack, Ph.D. St. John s University Tobin College of Business Queens, NY, 11439 pollacks@stjohns.edu

Name: University of Chicago Graduate School of Business Business 41000: Business Statistics Special Notes: 1. This is a closed-book exam. You may use an 8 11 piece of paper for the formulas. 2. Throughout

### Chapter 23 Inferences About Means

Chapter 23 Inferences About Means Chapter 23 - Inferences About Means 391 Chapter 23 Solutions to Class Examples 1. See Class Example 1. 2. We want to know if the mean battery lifespan exceeds the 300-minute

### 3.2 Roulette and Markov Chains

238 CHAPTER 3. DISCRETE DYNAMICAL SYSTEMS WITH MANY VARIABLES 3.2 Roulette and Markov Chains In this section we will be discussing an application of systems of recursion equations called Markov Chains.

### Good luck! BUSINESS STATISTICS FINAL EXAM INSTRUCTIONS. Name:

Glo bal Leadership M BA BUSINESS STATISTICS FINAL EXAM Name: INSTRUCTIONS 1. Do not open this exam until instructed to do so. 2. Be sure to fill in your name before starting the exam. 3. You have two hours

### p-values and significance levels (false positive or false alarm rates)

p-values and significance levels (false positive or false alarm rates) Let's say 123 people in the class toss a coin. Call it "Coin A." There are 65 heads. Then they toss another coin. Call it "Coin B."

### NCSS Statistical Software

Chapter 06 Introduction This procedure provides several reports for the comparison of two distributions, including confidence intervals for the difference in means, two-sample t-tests, the z-test, the

### V. RANDOM VARIABLES, PROBABILITY DISTRIBUTIONS, EXPECTED VALUE

V. RANDOM VARIABLES, PROBABILITY DISTRIBUTIONS, EXPETED VALUE A game of chance featured at an amusement park is played as follows: You pay \$ to play. A penny and a nickel are flipped. You win \$ if either

### Experimental Design. Power and Sample Size Determination. Proportions. Proportions. Confidence Interval for p. The Binomial Test

Experimental Design Power and Sample Size Determination Bret Hanlon and Bret Larget Department of Statistics University of Wisconsin Madison November 3 8, 2011 To this point in the semester, we have largely

### Lesson 1: Comparison of Population Means Part c: Comparison of Two- Means

Lesson : Comparison of Population Means Part c: Comparison of Two- Means Welcome to lesson c. This third lesson of lesson will discuss hypothesis testing for two independent means. Steps in Hypothesis

### 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

### 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,

### Discrete Mathematics and Probability Theory Fall 2009 Satish Rao, David Tse Note 13. Random Variables: Distribution and Expectation

CS 70 Discrete Mathematics and Probability Theory Fall 2009 Satish Rao, David Tse Note 3 Random Variables: Distribution and Expectation Random Variables Question: The homeworks of 20 students are collected

### Regression Analysis: A Complete Example

Regression Analysis: A Complete Example This section works out an example that includes all the topics we have discussed so far in this chapter. A complete example of regression analysis. PhotoDisc, Inc./Getty

### MOVIES, GAMBLING, SECRET CODES, JUST MATRIX MAGIC

MOVIES, GAMBLING, SECRET CODES, JUST MATRIX MAGIC DR. LESZEK GAWARECKI 1. The Cartesian Coordinate System In the Cartesian system points are defined by giving their coordinates. Plot the following points:

### STT315 Chapter 4 Random Variables & Probability Distributions KM. Chapter 4.5, 6, 8 Probability Distributions for Continuous Random Variables

Chapter 4.5, 6, 8 Probability Distributions for Continuous Random Variables Discrete vs. continuous random variables Examples of continuous distributions o Uniform o Exponential o Normal Recall: A random

### Two-Sample T-Tests Assuming Equal Variance (Enter Means)

Chapter 4 Two-Sample T-Tests Assuming Equal Variance (Enter Means) Introduction This procedure provides sample size and power calculations for one- or two-sided two-sample t-tests when the variances of

### Chapter 2 Probability Topics SPSS T tests

Chapter 2 Probability Topics SPSS T tests Data file used: gss.sav In the lecture about chapter 2, only the One-Sample T test has been explained. In this handout, we also give the SPSS methods to perform

### WISE Power Tutorial All Exercises

ame Date Class WISE Power Tutorial All Exercises Power: The B.E.A.. Mnemonic Four interrelated features of power can be summarized using BEA B Beta Error (Power = 1 Beta Error): Beta error (or Type II

### LAB 4 INSTRUCTIONS CONFIDENCE INTERVALS AND HYPOTHESIS TESTING

LAB 4 INSTRUCTIONS CONFIDENCE INTERVALS AND HYPOTHESIS TESTING In this lab you will explore the concept of a confidence interval and hypothesis testing through a simulation problem in engineering setting.

Name: OUTLINE SOLUTIONS University of Chicago Graduate School of Business Business 41000: Business Statistics Solution Key Special Notes: 1. This is a closed-book exam. You may use an 8 11 piece of paper

### Hypothesis Test for Mean Using Given Data (Standard Deviation Known-z-test)

Hypothesis Test for Mean Using Given Data (Standard Deviation Known-z-test) A hypothesis test is conducted when trying to find out if a claim is true or not. And if the claim is true, is it significant.

### 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

### Tutorial 5: Hypothesis Testing

Tutorial 5: Hypothesis Testing Rob Nicholls nicholls@mrc-lmb.cam.ac.uk MRC LMB Statistics Course 2014 Contents 1 Introduction................................ 1 2 Testing distributional assumptions....................

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

Sample Practice problems - chapter 12-1 and 2 proportions for inference - Z Distributions Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Provide

### Expected Value. 24 February 2014. Expected Value 24 February 2014 1/19

Expected Value 24 February 2014 Expected Value 24 February 2014 1/19 This week we discuss the notion of expected value and how it applies to probability situations, including the various New Mexico Lottery

### THE WINNING ROULETTE SYSTEM by http://www.webgoldminer.com/

THE WINNING ROULETTE SYSTEM by http://www.webgoldminer.com/ Is it possible to earn money from online gambling? Are there any 100% sure winning roulette systems? Are there actually people who make a living

### Example. A casino offers the following bets (the fairest bets in the casino!) 1 You get \$0 (i.e., you can walk away)

: Three bets Math 45 Introduction to Probability Lecture 5 Kenneth Harris aharri@umich.edu Department of Mathematics University of Michigan February, 009. A casino offers the following bets (the fairest

### Two-Sample T-Tests Allowing Unequal Variance (Enter Difference)

Chapter 45 Two-Sample T-Tests Allowing Unequal Variance (Enter Difference) Introduction This procedure provides sample size and power calculations for one- or two-sided two-sample t-tests when no assumption

### Chapter 8 Section 1. Homework A

Chapter 8 Section 1 Homework A 8.7 Can we use the large-sample confidence interval? In each of the following circumstances state whether you would use the large-sample confidence interval. The variable

### Chicago Booth BUSINESS STATISTICS 41000 Final Exam Fall 2011

Chicago Booth BUSINESS STATISTICS 41000 Final Exam Fall 2011 Name: Section: I pledge my honor that I have not violated the Honor Code Signature: This exam has 34 pages. You have 3 hours to complete this

### Non-Inferiority Tests for One Mean

Chapter 45 Non-Inferiority ests for One Mean Introduction his module computes power and sample size for non-inferiority tests in one-sample designs in which the outcome is distributed as a normal random

### The Variability of P-Values. Summary

The Variability of P-Values Dennis D. Boos Department of Statistics North Carolina State University Raleigh, NC 27695-8203 boos@stat.ncsu.edu August 15, 2009 NC State Statistics Departement Tech Report

### An introduction to statistical data analysis (Summer 2014) Lecture notes. Taught by Shravan Vasishth [vasishth@uni-potsdam.de]

An introduction to statistical data analysis (Summer 2014) Lecture notes Taught by Shravan Vasishth [vasishth@uni-potsdam.de] Last edited: May 9, 2014 2 > sessioninfo() R version 3.0.2 (2013-09-25) Platform:

### QUANTITATIVE METHODS BIOLOGY FINAL HONOUR SCHOOL NON-PARAMETRIC TESTS

QUANTITATIVE METHODS BIOLOGY FINAL HONOUR SCHOOL NON-PARAMETRIC TESTS This booklet contains lecture notes for the nonparametric work in the QM course. This booklet may be online at http://users.ox.ac.uk/~grafen/qmnotes/index.html.

### Testing a claim about a population mean

Introductory Statistics Lectures Testing a claim about a population mean One sample hypothesis test of the mean Department of Mathematics Pima Community College Redistribution of this material is prohibited

### Betting systems: how not to lose your money gambling

Betting systems: how not to lose your money gambling G. Berkolaiko Department of Mathematics Texas A&M University 28 April 2007 / Mini Fair, Math Awareness Month 2007 Gambling and Games of Chance Simple

### General Method: Difference of Means. 3. Calculate df: either Welch-Satterthwaite formula or simpler df = min(n 1, n 2 ) 1.

General Method: Difference of Means 1. Calculate x 1, x 2, SE 1, SE 2. 2. Combined SE = SE1 2 + SE2 2. ASSUMES INDEPENDENT SAMPLES. 3. Calculate df: either Welch-Satterthwaite formula or simpler df = min(n

### Probability: The Study of Randomness Randomness and Probability Models. IPS Chapters 4 Sections 4.1 4.2

Probability: The Study of Randomness Randomness and Probability Models IPS Chapters 4 Sections 4.1 4.2 Chapter 4 Overview Key Concepts Random Experiment/Process Sample Space Events Probability Models Probability

### 36 Odds, Expected Value, and Conditional Probability

36 Odds, Expected Value, and Conditional Probability What s the difference between probabilities and odds? To answer this question, let s consider a game that involves rolling a die. If one gets the face

### Fairfield Public Schools

Mathematics Fairfield Public Schools AP Statistics AP Statistics BOE Approved 04/08/2014 1 AP STATISTICS Critical Areas of Focus AP Statistics is a rigorous course that offers advanced students an opportunity

### Term Project: Roulette

Term Project: Roulette DCY Student January 13, 2006 1. Introduction The roulette is a popular gambling game found in all major casinos. In contrast to many other gambling games such as black jack, poker,

### Module 4 (Effect of Alcohol on Worms): Data Analysis

Module 4 (Effect of Alcohol on Worms): Data Analysis Michael Dunn Capuchino High School Introduction In this exercise, you will first process the timelapse data you collected. Then, you will cull (remove)

### Calculating P-Values. Parkland College. Isela Guerra Parkland College. Recommended Citation

Parkland College A with Honors Projects Honors Program 2014 Calculating P-Values Isela Guerra Parkland College Recommended Citation Guerra, Isela, "Calculating P-Values" (2014). A with Honors Projects.

### Probability and Expected Value

Probability and Expected Value This handout provides an introduction to probability and expected value. Some of you may already be familiar with some of these topics. Probability and expected value are

### Hypothesis Testing. Steps for a hypothesis test:

Hypothesis Testing Steps for a hypothesis test: 1. State the claim H 0 and the alternative, H a 2. Choose a significance level or use the given one. 3. Draw the sampling distribution based on the assumption

### Stat 411/511 THE RANDOMIZATION TEST. Charlotte Wickham. stat511.cwick.co.nz. Oct 16 2015

Stat 411/511 THE RANDOMIZATION TEST Oct 16 2015 Charlotte Wickham stat511.cwick.co.nz Today Review randomization model Conduct randomization test What about CIs? Using a t-distribution as an approximation

### Choice Under Uncertainty

Decision Making Under Uncertainty Choice Under Uncertainty Econ 422: Investment, Capital & Finance University of ashington Summer 2006 August 15, 2006 Course Chronology: 1. Intertemporal Choice: Exchange

### 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!

### Section 13, Part 1 ANOVA. Analysis Of Variance

Section 13, Part 1 ANOVA Analysis Of Variance Course Overview So far in this course we ve covered: Descriptive statistics Summary statistics Tables and Graphs Probability Probability Rules Probability

### Chapter 7 Section 1 Homework Set A

Chapter 7 Section 1 Homework Set A 7.15 Finding the critical value t *. What critical value t * from Table D (use software, go to the web and type t distribution applet) should be used to calculate the

### NONPARAMETRIC STATISTICS 1. depend on assumptions about the underlying distribution of the data (or on the Central Limit Theorem)

NONPARAMETRIC STATISTICS 1 PREVIOUSLY parametric statistics in estimation and hypothesis testing... construction of confidence intervals computing of p-values classical significance testing depend on assumptions

### Outline. Definitions Descriptive vs. Inferential Statistics The t-test - One-sample t-test

The t-test Outline Definitions Descriptive vs. Inferential Statistics The t-test - One-sample t-test - Dependent (related) groups t-test - Independent (unrelated) groups t-test Comparing means Correlation

### Chapter 5. Discrete Probability Distributions

Chapter 5. Discrete Probability Distributions Chapter Problem: Did Mendel s result from plant hybridization experiments contradicts his theory? 1. Mendel s theory says that when there are two inheritable

### Data Mining Techniques Chapter 5: The Lure of Statistics: Data Mining Using Familiar Tools

Data Mining Techniques Chapter 5: The Lure of Statistics: Data Mining Using Familiar Tools Occam s razor.......................................................... 2 A look at data I.........................................................

### Some Essential Statistics The Lure of Statistics

Some Essential Statistics The Lure of Statistics Data Mining Techniques, by M.J.A. Berry and G.S Linoff, 2004 Statistics vs. Data Mining..lie, damn lie, and statistics mining data to support preconceived

### In the past, the increase in the price of gasoline could be attributed to major national or global

Chapter 7 Testing Hypotheses Chapter Learning Objectives Understanding the assumptions of statistical hypothesis testing Defining and applying the components in hypothesis testing: the research and null

### Math 108 Exam 3 Solutions Spring 00

Math 108 Exam 3 Solutions Spring 00 1. An ecologist studying acid rain takes measurements of the ph in 12 randomly selected Adirondack lakes. The results are as follows: 3.0 6.5 5.0 4.2 5.5 4.7 3.4 6.8

### Betting on Excel to enliven the teaching of probability

Betting on Excel to enliven the teaching of probability Stephen R. Clarke School of Mathematical Sciences Swinburne University of Technology Abstract The study of probability has its roots in gambling

### Hypothesis Tests for 1 sample Proportions

Hypothesis Tests for 1 sample Proportions 1. Hypotheses. Write the null and alternative hypotheses you would use to test each of the following situations. a) A governor is concerned about his "negatives"

### Example: Find the expected value of the random variable X. X 2 4 6 7 P(X) 0.3 0.2 0.1 0.4

MATH 110 Test Three Outline of Test Material EXPECTED VALUE (8.5) Super easy ones (when the PDF is already given to you as a table and all you need to do is multiply down the columns and add across) Example:

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

Final Exam Review MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 1) A researcher for an airline interviews all of the passengers on five randomly

### HYPOTHESIS TESTING WITH SPSS:

HYPOTHESIS TESTING WITH SPSS: A NON-STATISTICIAN S GUIDE & TUTORIAL by Dr. Jim Mirabella SPSS 14.0 screenshots reprinted with permission from SPSS Inc. Published June 2006 Copyright Dr. Jim Mirabella CHAPTER

### MTH 140 Statistics Videos

MTH 140 Statistics Videos Chapter 1 Picturing Distributions with Graphs Individuals and Variables Categorical Variables: Pie Charts and Bar Graphs Categorical Variables: Pie Charts and Bar Graphs Quantitative

### 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