1 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 error) is the probability that a test of statistical significance will fail to reject the null hypothesis when it is false (e.g., when there really is an effect of training). As beta error increases, power decreases. E Effect Size: The effect size is the magnitude of the difference between the actual population mean and the null hypothesized mean (µ 1 µ 0 ) relative to standard deviation of scores (σ). When the effect size d = (µ 1 µ 0 ) / σ in the population is larger, the null and population sampling distributions overlap less and power is greater. As effect size increases, power increases (assuming no change in alpha or sample size). A Alpha error: Alpha error (or Type I error) is the probability that a statistical test will produce a statistically significant finding when the null hypothesis is true (e.g., there is no effect of training). For example, if alpha error =.05 and the null hypothesis is true, then out of 100 statistical tests, false significance would be found on average 5 times. The risk of false significance would be 5%. In practice, alpha is typically set by the researcher at.01 or.05 As alpha error increases, power increases (assuming no change in effect size or sample size). Sample Size: As the sample size increases, the variability of sample means decreases. The population and null sampling distributions become narrower, overlapping to a lesser extent and making it easier to detect a difference between these distributions. This results in greater power. As sample size increases, power increases (assuming no change in alpha or effect size).
2 Select true or false for each scenario: (Assuming no other changes) 1. As effect size increases, power decreases. 2. As sample size increases, power increases. 3. As alpha error increases, power decreases. 4. Beta error is unrelated to power. True False 1. Power and Effect Size As the effect size increases, the power of a statistical test increases. The effect size, d, is defined as the number of standard deviations between the null mean and the alternate mean. Symbolically, where d is the effect size, µ 0 is the population mean for the null distribution, µ 1 is the population mean for the alternative distribution, and σ is the standard deviation for both the null and alternative distributions. From the equation it can be noted that two factors impact the effect size: 1) the difference between the null and alternative distribution means, and 2) the standard deviation. For any given population standard deviation, the greater the difference between the means of the null and alternative distributions, the greater the power. Further, for any given difference in means, power is greater if the standard deviation is smaller. In the following exercise, we will use the power applet to explore how the effect size influences power. Your task is to find a good way to explain how this works to a friend. Exercise 1a: Power and Mean Differences (Large Effect) How probable is it that a sample of graduates from the ACE training program will provide convincing statistical evidence that ACE graduates perform better than non-graduates on the standardized Verbal Ability and Skills Test (VAST)? How likely is it that a rival competitor, the DEUCE training program, will provide convincing evidence? Power analysis will allow us to answer these questions. We will use the WISE Power Applet to examine and compare the statistical power of our tests to detect the claims of the ACE and DEUCE training programs. We begin with a test of ACE graduates.
3 We assume that for the population of non-graduates of a training course, the mean on VAST is 500 with a standard deviation of 100. For the population of ACE graduates the mean is 580 and the standard deviation is 100. Symbolically, µ 0 = 500, µ 1 = 580, and σ = 100. Both distributions are assumed to be normal. How large is the effect size? The formula for d shown below indicates that the effect size for the ACE program is.80. This tells us that the mean for the alternative population is.80 standard deviations greater than the mean for the null population. The z-score of a sample mean computed on the null sampling distribution allows us to determine the probability of observing a sample mean this large or larger if the null hypothesis is true. To prepare the simulation, enter the following information into the applet below: µ 0 = 500 (null mean); µ 1 = 580 (alternative mean); σ = 100 (standard deviation); α =.05 (alpha error rate, one tailed); n = 25 (sample size). To simulate drawing one sample of 25 cases, press Sample. The mean and z-score are shown in the applet (bottom right box). Record these values in the first pair of boxes below (you may round the mean to a whole number). Trial Mean Z-Score Is this sample mean large enough to allow you to reject the null hypothesis? How likely is it that you would observe a sample this large or larger if the null hypothesis was true so that you really were sampling from the blue distribution? (Answer: The p-value is the probability of observing a mean as large or larger than your sample mean if the null hypothesis is true.) Now draw two more samples and record the mean and z for each in the boxes. These values will be saved and used later and can be printed for a homework exercise. Some of the boxes have already been filled out for you.
4 The power of this statistical test is the probability that the mean of a random sample of size n will be large enough to allow us to correctly reject the null hypothesis. Because we are actually sampling from the Alternative Population (red distribution), the probability that we will observe a sample mean large enough to reject H 0 corresponds to the proportion of the red sampling distribution that is to the right of the dashed line. For this example, we can use the value provided by the applet,.991. Thus, if we draw a sample of 25 cases from ACE graduates, the probability is 99.1% that our sample mean will be large enough that we can reject the null hypothesis that the sample came from a population with a mean of only 500. The probability that we will fail to reject H 0 is only =.009, less than one chance in a. How many times could you reject the null hypothesis in your ten samples? (With one-tailed alpha α =.05, z = 1.645, so reject H 0 if your z-score is greater than 1.645) Exercise 1b: Power and Mean Differences (Small Effect) Now we will assess the power of a test for a rival training program, the DEUCE program. The mean score for the population of graduates of this program is 520. Again we assume the population distribution is normal with a standard deviation of 100. Using the formula for d, we find that the population effect size for the DEUCE program is only.20. Recall the effect size for the ACE program was much larger: 1b. Before drawing samples, consider how the statistical power will differ for a test of DEUCE graduates compared to the power we found for a test of ACE graduates. That is, do you expect you will be more likely or less likely to reject the null hypotheses for a sample of 25 graduates drawn from the DEUCE program compared to a similar test for the ACE program? I predict that statistical power for the test of the DEUCE program compared to the test of the ACE program will be: Less The Same Greater
5 With the applet you will be able to change the effect size and watch what happens to statistical power. Your goal for this exercise is to be able to explain to a friend why statistical power is greater when the effect size is greater. To simulate drawing a sample of 25 from graduates from the DEUCE program, enter the following information into the WISE Power Applet: µ 0 = 500 (null mean); µ 1 = 520 (alternative mean); σ = 100 (standard deviation); α =.05 (alpha error rate, one tailed); n = 25 (sample size). Do three simulations of drawing a sample of 25 cases, and record the results below. Trial Mean Z-Score c. What is the power for this test as shown in the applet? 1d. How many of your ten simulated samples allowed you to reject the null hypothesis? (Use one-tailed alpha α =.05, z = 1.645, so reject H 0 if your z-score is greater than 1.645) 1e. For the ACE program, the effect size was.8 and the power of the statistical test was.991; what can you conclude about the relationship between effect size and power? A. The test for the ACE program, which had a larger effect size, had more power. B. The test for the DEUCE program, which had a smaller effect size, had more power. C. Effect size is unrelated to power.
6 Exercise 1c: Power and Variability (Standard Deviation) In Exercises 1a and 1b, we examined how differences between the means of the null and alternative populations affect power. In this exercise, we will investigate another variable that impacts the effect size and power; the variability of the population. If the standard deviation for graduates of the TREY program was only 50 instead of 100, do you think power would be greater or less than for the DEUCE program (assume the population means are 520 for graduates of both programs)? Think about what will happen before you try the simulation. Referencing the effect size calculation may help you formulate your opinion: 1f.I think that with a smaller standard deviation in the population, the statistical power will be Less Unchanged Greater I don't know To simulate drawing a sample from graduates of the TREY program that has the same population mean as the DEUCE program (520), but a smaller standard deviation (50 instead of 100), enter the following values into the WISE Power Applet: µ 0 = 500 (null mean); µ 1 = 520 (alternative mean); σ = 50 (standard deviation); α =.05 (alpha error rate, one tailed); n = 25 (sample size). Do three simulations of drawing a sample of 25 cases and record the results below. Trial Mean Z-Score g. How many of your ten simulated samples allowed you to reject the null hypothesis? (Use one-tailed alpha α =.05, z = 1.645, so reject H 0 if your z-score is greater than 1.645) 1h. What is the power for this test (from the applet)?
7 1i. In Exercise 1b the DEUCE program had a mean of 520 just like the TREY program, but with samples of N = 25 for both programs, the test for the DEUCE program had a power of.260 rather than.639. The standard deviation for DEUCE was 100 rather than 50. Why is statistical power greater for the TREY program? Because smaller population variance always produces greater power. Because the program with the larger effect size always produces greater power. Neither of these reasons is sufficient. Exercise 1d: Summary of Power and Effect Size The formula for effect size is. We can see from the formula that two variables impact the effect size: the difference between the means of the alternative and null populations and the standard deviation of the two populations. Below are two images that may be useful for examining the impact of standard deviation and mean differences on power. Same Standard Deviation, Different Means When the difference between the means of the null and the alternative distributions is increased, the sampling distributions do not change shape, but they are further apart on the x-axis. Note that when the difference is increased, the sampling distributions overlap to a lesser extent, and it is less likely that a random sample from the alternative distribution will be mistaken for a sample taken from the null distribution. More of the sampling distribution for the alternate population (the pink distribution) is greater than the critical value (red dashed line), so it is more likely that a random sample mean will lead to rejection of the null hypothesis.
8 Same Mean Difference, Different Standard Deviation When the standard deviation is smaller, the sampling distributions are narrower, but the means remain the same distance apart on the x-axis. The sampling distributions overlap to a lesser extent because they are narrower. Compare the proportion of the alternate (red) distribution that is above or below the critical value in the two scenarios. The following questions are designed to test your understanding of the factors that affect power. Below are key statistics for each of two new training programs, SLAM and DUNK. Do you expect that a test of statistical significance would have greater power for the SLAM program or the DUNK program? Why? Respond to the following true/false statements. See if you can answer all statements correctly before you check your answers. T F 1j. The test of the SLAM program will have greater power because SLAM has a larger mean than DUNK. 1k. The test of the SLAM program will have greater power because SLAM has a larger standard deviation than DUNK. 1l. The test of the DUNK program will have greater power, because a program with smaller standard deviation has greater power.
9 1m. The test of the SLAM program will have greater power because SLAM has a greater effect size. 1n. The test of the DUNK program will have greater power than the test for the SLAM program if sample size and alpha are the same, because DUNK has a greater effect size. 1o. Power is the same for the tests of the two programs because the samples have the same size. 1p. Holding all other factors constant, a larger difference between the null and alternate population means will always yield greater power. 1q.Holding all other factors constant, power is greater when the variance of the null and alternate populations is greater. Exercise 2: Power and Sample Size Why does statistical power increase as sample size increases (assuming effect size and alpha are unchanged)? The goal of this exercise is to help you develop an explanation that you could give to a classmate. In Exercise 1d we drew samples of 25 graduates from the DEUCE program, but in Exercise 2 we will draw samples of n = 100 and n = 4. Watch what happens and think about how you can explain how statistical power is influenced by sample size. To simulate drawing a larger sample, enter the following values into the WISE Power Applet: µ 0 = 500 (null mean); µ 1 = 520 (alternative mean); σ = 100 (standard deviation); α =.05 (alpha error rate, one tailed); n = 100 (sample size). For this exercise, do ten simulations of drawing a sample of 100 cases and record the results below. You don't need to record means; just select the button for "Reject H 0 " or "Fail to Reject" for each of the ten simulations.
10 First, do ten simulations of drawing a sample of 100 cases and record the results below Reject H 0 Fail to Reject 2a. Power (as shown in the applet) = Next, do ten simulations of drawing a sample of 4 cases and record the results below. (Set sample size n = 4, press Enter) Reject H 0 Fail to Reject 2b. Power (as shown in the applet) = 2c. How many times out of 10 did you Reject H 0 for each of the two scenarios? n = 4: n = 100: If nothing else is changed, power is greater with a larger sample size because: (select True or False for each before you check your answers): T F 2d. The effect size is larger. 2e. The alpha error is smaller. 2f. The alpha error is larger. 2g. The population variance is smaller. 2h. The variance of the alternate sampling distribution is smaller. 2i. More of the alternate sampling distribution exceeds the critical value.
11 We have seen that the power of a statistical test increases as the sample increases. Let's review the reasons behind this increase in power. You may have noticed that the sampling distribution of the mean was much wider when the sample size was reduced from 100 to 4. This occurred because sample means are much more variable when the sample size is small. Regardless of sample size, the average sample mean, taken across infinite samples, would be equal to the population mean. However, the precision of sample means is less when the sample size is small. This can be observed as the greater variability in sampling distributions for the sample size of 4. When the sampling distributions are more variable, it is more likely that a sample taken from the alternative distribution will be mistaken for a sample taken from the null distribution. This mistake leads to failure to reject the null hypothesis, which corresponds to lower power. The figures below show sampling distributions for samples of 100 compared to samples of 4. Although the difference between the means is the same, there is much less overlap of the sampling distributions for the larger samples. 2j. Explain to a classmate why statistical power increases as the sample size increases.
12 Exercise 3: Power and Alpha Now, we will consider the impact of using a different alpha value, α. As the researcher, we decide on the value of alpha, typically at.05 or.01. Alpha is the error rate we are willing to accept for the error of rejecting the null hypothesis if it were true. We require stronger evidence to reject the null hypothesis if we set alpha at.01 than if we use alpha of.05. The figures show how power and alpha error are related. When we reduce alpha error (represented by the dark blue area of the null distribution to the right of the red dashed line), we also reduce power (the pink area in the alternate distribution to the right of the dashed line). For this example, use one-tailed alpha α =.01 (z = 2.326). In this case, we will reject the null hypothesis only if a sample mean is so large that it would occur less than 1% of the time given the null hypothesis is true. You do not need to draw additional samples for this problem; you can use the data recorded for samples drawn in Exercise 1 (µ 0 = 500, σ = 100, n = 25, α =.05, z = 1.645). Data from Exercise 1 ACE Program (µ 1 = 580) Trial Mean Z-Score DEUCE Program (µ 1 = 520) Trial Mean Z-Score
13 3a. Using alpha of.01 instead of.05, how many times could you reject the null hypothesis for your results in Exercise 1? (How many times is Z > 2.326?) Reject for ACE Program (µ 1 = 580) Reject for DEUCE Program (µ 1 = 520) α =.05 (from #1) α =.01 3b. What is the power for each of these tests? You can use the applet below to calculate power for the tests using alpha α =.01. (Set n = 25 and µ 0 = 500 for all tests ; use µ 1 = 580 for ACE and µ 1 = 520 for DEUCE). Remember to press 'Enter' after each change to the applet. Power for ACE Program (µ 1 = 580) Power for DEUCE Program (µ 1 = 520) α =.05 (from #1) α =
14 Cumulative Test: What affects Statistical Power? Select True or False for each of the following questions. If nothing else is changed, power is greater when T F C1. The difference between the null and alternative population means is greater. C2. The standard deviation of the populations is greater. C3. The alpha error rate is changed from.01 to.05. C4. The sample size is changed from 30 to 40. More challenging questions: T F C5. Power is always greater when the effect size is greater. C6. Power is always greater when the Type II error (i.e., beta error) is smaller. C7. To compute power, all I need to know is effect size, sample size, and alpha. C8. Which of the following situations would yield the greatest power (assuming alpha and sample size are held constant)? Null mean = 500, Alternative mean = 510, Standard Deviation = 40 Null mean = 500, Alternative mean = 540, Standard Deviation = 160 Null mean = 500, Alternative mean = 520, Standard Deviation = 60 C9. Consider the shape of the sampling distributions for samples of size n = 4, n = 25, and n = 100. What happens to the sampling distribution of the sample mean when n is increased (assuming nothing else changes)? Sampling distribution becomes more disperse. Sampling distribution becomes less disperse. Sampling distribution remains the same.
15 C10. So far you have examined the effect of magnitude of difference between the null mean and the alternative mean, standard deviation, sample size, and alpha level on power. Which of the answers below best summarizes the effect of each on power? More power = large magnitude of difference, larger standard deviation, larger sample, larger alpha. More power = large magnitude of difference, smaller standard deviation, larger sample, smaller alpha. More power = large magnitude of difference, smaller standard deviation, larger sample, larger alpha. More power = smaller magnitude of difference, smaller standard deviation, larger sample, smaller alpha.
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
Section 7.1 Introduction to Hypothesis Testing Schrodinger s cat quantum mechanics thought experiment (1935) Statistical Hypotheses A statistical hypothesis is a claim about a population. Null hypothesis
Introduction to Hypothesis Testing 1 Hypothesis Testing A hypothesis test is a statistical procedure that uses sample data to evaluate a hypothesis about a population Hypothesis is stated in terms of the
Population and sample Sampling and Hypothesis Testing Allin Cottrell Population : an entire set of objects or units of observation of one sort or another. Sample : subset of a population. Parameter versus
I. Terms, Concepts. Introduction to Hypothesis Testing A. In general, we do not know the true value of population parameters - they must be estimated. However, we do have hypotheses about what the true
HYPOTHESIS TESTING: POWER OF THE TEST The first 6 steps of the 9-step test of hypothesis are called "the test". These steps are not dependent on the observed data values. When planning a research project,
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. email@example.com www.excelmasterseries.com
THE LOGIC OF HYPOTHESIS TESTING Hypothesis testing is a statistical procedure that allows researchers to use sample to draw inferences about the population of interest. It is the most commonly used inferential
Chapter 8 Hypothesis Testing 1 Chapter 8 Hypothesis Testing 8-1 Overview 8-2 Basics of Hypothesis Testing 8-3 Testing a Claim About a Proportion 8-5 Testing a Claim About a Mean: s Not Known 8-6 Testing
Hypothesis Testing Summary Hypothesis testing begins with the drawing of a sample and calculating its characteristics (aka, statistics ). A statistical test (a specific form of a hypothesis test) is an
DDBA 8438: Introduction to Hypothesis Testing Video Podcast Transcript JENNIFER ANN MORROW: Welcome to "Introduction to Hypothesis Testing." My name is Dr. Jennifer Ann Morrow. In today's demonstration,
Inferential Statistics Sampling and the normal distribution Z-scores Confidence levels and intervals Hypothesis testing Commonly used statistical methods Inferential Statistics Descriptive statistics are
Module 5 Hypotheses Tests: Comparing Two Groups Objective: In medical research, we often compare the outcomes between two groups of patients, namely exposed and unexposed groups. At the completion of this
AP STATISTICS 2009 SCORING GUIDELINES (Form B) Question 5 Intent of Question The primary goals of this question were to assess students ability to (1) state the appropriate hypotheses, (2) identify and
HOW TO WRITE A LABORATORY REPORT Pete Bibby Dept of Psychology 1 About Laboratory Reports The writing of laboratory reports is an essential part of the practical course One function of this course is to
Open book and note Calculator OK Multiple Choice 1 point each MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the mean for the given sample data.
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.
Correlational Research Chapter Fifteen Correlational Research Chapter Fifteen Bring folder of readings The Nature of Correlational Research Correlational Research is also known as Associational Research.
Chapter 8 Hypothesis Testing Hypothesis In statistics, a hypothesis is a claim or statement about a property of a population. A hypothesis test (or test of significance) is a standard procedure for testing
MINITAB ASSISTANT WHITE PAPER This paper explains the research conducted by Minitab statisticians to develop the methods and data checks used in the Assistant in Minitab 17 Statistical Software. One-Way
Hypothesis test In statistics, a hypothesis is a claim or statement about a property of a population. A hypothesis test (or test of significance) is a standard procedure for testing a claim about a property
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
Section 7.1 - Introduction to Hypothesis Testing Chapter 7 Objectives: State a null hypothesis and an alternative hypothesis Identify type I and type II errors and interpret the level of significance Determine
Descriptive Statistics Primer Descriptive statistics Central tendency Variation Relative position Relationships Calculating descriptive statistics Descriptive Statistics Purpose to describe or summarize
About Hypothesis Testing TABLE OF CONTENTS About Hypothesis Testing... 1 What is a HYPOTHESIS TEST?... 1 Hypothesis Testing... 1 Hypothesis Testing... 1 Steps in Hypothesis Testing... 2 Steps in Hypothesis
Chapter 9 Two-Sample Tests Paired t Test (Correlated Groups t Test) Effect Sizes and Power Paired t Test Calculation Summary Independent t Test Chapter 9 Homework Power and Two-Sample Tests: Paired Versus
Sampling Distribution of the Mean & Hypothesis Testing Let s first review what we know about sampling distributions of the mean (Central Limit Theorem): 1. The mean of the sampling distribution will be
CHAPTER 11 SECTION 2: INTRODUCTION TO HYPOTHESIS TESTING MULTIPLE CHOICE 56. In testing the hypotheses H 0 : µ = 50 vs. H 1 : µ 50, the following information is known: n = 64, = 53.5, and σ = 10. The standardized
CHANCE ENCOUNTERS Making Sense of Hypothesis Tests Howard Fincher Learning Development Tutor Upgrade Study Advice Service Oxford Brookes University Howard Fincher 2008 PREFACE This guide has a restricted
8-2 Basics of Hypothesis Testing Definitions This section presents individual components of a hypothesis test. We should know and understand the following: How to identify the null hypothesis and alternative
Hypothesis Testing or How to Decide to Decide Edpsy 580 Carolyn J. Anderson Department of Educational Psychology University of Illinois at Urbana-Champaign Hypothesis Testing or How to Decide to Decide
CHAPTER 8 Learning Objectives C H A P T E R E I G H T Hypothesis Testing 1 Outline 8-1 Steps in Traditional Method 8-2 z Test for a Mean 8-3 t Test for a Mean 8-4 z Test for a Proportion 8-5 2 Test for
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
1 14.1 Using the Binomial Table Nonparametric Statistics In this chapter, we will survey several methods of inference from Nonparametric Statistics. These methods will introduce us to several new tables
Chapter 8: Introduction to Hypothesis Testing We re now at the point where we can discuss the logic of hypothesis testing. This procedure will underlie the statistical analyses that we ll use for the remainder
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
Chapter 1 Hypothesis Testing Principles of Hypothesis Testing tests for one sample case 1 Statistical Hypotheses They are defined as assertion or conjecture about the parameter or parameters of a population,
Lecture 13 More on hypothesis testing Thais Paiva STA 111 - Summer 2013 Term II July 22, 2013 1 / 27 Thais Paiva STA 111 - Summer 2013 Term II Lecture 13, 07/22/2013 Lecture Plan 1 Type I and type II error
Notation: Notation and Equations for Regression Lecture 11/4 m: The number of predictor variables in a regression Xi: One of multiple predictor variables. The subscript i represents any number from 1 through
PSY 511: Advanced Statistics for Psychological and Behavioral Research 1 All inferential statistics have the following in common: Use of some descriptive statistic Use of probability Potential for estimation
HYPOTHESIS TESTING (ONE SAMPLE) - CHAPTER 7 1 PREVIOUSLY used confidence intervals to answer questions such as... You know that 0.25% of women have red/green color blindness. You conduct a study of men
Chapter 9, Part A Hypothesis Tests Slide 1 Learning objectives 1. Understand how to develop Null and Alternative Hypotheses 2. Understand Type I and Type II Errors 3. Able to do hypothesis test about population
Textbook Reference: Chapter 9.5 Measuring the Power of a Test An economic problem motivates the statement of a null and alternative hypothesis. For a numeric data set, a decision rule can lead to the rejection
Chapter 640 Confidence Intervals for One Standard Deviation Using Standard Deviation Introduction This routine calculates the sample size necessary to achieve a specified interval width or distance from
C H 8A P T E R Outline 8 1 Steps in Traditional Method 8 2 z Test for a Mean 8 3 t Test for a Mean 8 4 z Test for a Proportion 8 6 Confidence Intervals and Copyright 2013 The McGraw Hill Companies, Inc.
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
1 Central Tendency CENTRAL TENDENCY: A statistical measure that identifies a single score that is most typical or representative of the entire group Usually, a value that reflects the middle of the distribution
Chapter 8 Student Lecture Notes 8-1 Chapter 8 Introduction to Hypothesis Testing Fall 26 Fundamentals of Business Statistics 1 Chapter Goals After completing this chapter, you should be able to: Formulate
1 Hypothesis Testing In Statistics, a hypothesis proposes a model for the world. Then we look at the data. If the data are consistent with that model, we have no reason to disbelieve the hypothesis. Data
HYPOTHESIS TESTING AND TYPE I AND TYPE II ERROR Hypothesis is a conjecture (an inferring) about one or more population parameters. Null Hypothesis (H 0 ) is a statement of no difference or no relationship
Chapter 10 Hypothesis Testing: Two Means, Paired Data, Two Proportions 10.1 Hypothesis Testing: Two Population Means and Two Population Proportions 1 10.1.1 Student Learning Objectives By the end of this
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
Chapter III Testing Hypotheses R (Introduction) A statistical hypothesis is an assumption about a population parameter This assumption may or may not be true The best way to determine whether a statistical
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
Introduction to Hypothesis Testing OPRE 6301 Motivation... The purpose of hypothesis testing is to determine whether there is enough statistical evidence in favor of a certain belief, or hypothesis, about
Module 2 Probability and Statistics BASIC CONCEPTS Multiple Choice Identify the choice that best completes the statement or answers the question. 1. The standard deviation of a standard normal distribution
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
SPSS on two independent samples. Two sample test with proportions. Paired t-test (with more SPSS) State of the course address: The Final exam is Aug 9, 3:30pm 6:30pm in B9201 in the Burnaby Campus. (One
Reasoning with Uncertainty More about Hypothesis Testing P-values, types of errors, power of a test P-Values and Decisions Your conclusion about any null hypothesis should be accompanied by the P-value
Chapter 807 Point Biserial Correlation Tests Introduction The point biserial correlation coefficient (ρ in this chapter) is the product-moment correlation calculated between a continuous random variable
Biodiversity Data Analysis: Testing Statistical Hypotheses By Joanna Weremijewicz, Simeon Yurek, Steven Green, Ph. D. and Dana Krempels, Ph. D. In biological science, investigators often collect biological
Contents 6.4 Normal Distribution....................... 381 6.4.1 Characteristics of the Normal Distribution....... 381 6.4.2 The Standardized Normal Distribution......... 385 6.4.3 Meaning of Areas under
Lecture 6: Chapter 6: Normal Probability Distributions A normal distribution is a continuous probability distribution for a random variable x. The graph of a normal distribution is called the normal curve.
How to Conduct a Hypothesis Test The idea of hypothesis testing is relatively straightforward. In various studies we observe certain events. We must ask, is the event due to chance alone, or is there some
Chapter 250 Introduction The Chi-square test is often used to test whether sets of frequencies or proportions follow certain patterns. The two most common instances are tests of goodness of fit using multinomial
HYPOTHESIS TESTING (ONE SAMPLE) - CHAPTER 7 1 PREVIOUSLY used confidence intervals to answer questions such as... You know that 0.25% of women have red/green color blindness. You conduct a study of men
Review #2 Statistics Find the mean of the given probability distribution. 1) x P(x) 0 0.19 1 0.37 2 0.16 3 0.26 4 0.02 A) 1.64 B) 1.45 C) 1.55 D) 1.74 2) The number of golf balls ordered by customers of
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
Statistical Inference Idea: Estimate parameters of the population distribution using data. How: Use the sampling distribution of sample statistics and methods based on what would happen if we used this
AP Statistics 2014 Scoring Guidelines College Board, Advanced Placement Program, AP, AP Central, and the acorn logo are registered trademarks of the College Board. AP Central is the official online home
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
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 450 on-inferiority Tests for Two Means using Differences Introduction This procedure computes power and sample size for non-inferiority tests in two-sample designs in which the outcome is a continuous
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
CHAPTER 5. A POPULATION MEAN, CONFIDENCE INTERVALS AND HYPOTHESIS TESTING 5.1 Concepts When a number of animals or plots are exposed to a certain treatment, we usually estimate the effect of the treatment
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.
Introduction to Hypothesis Testing Point estimation and confidence intervals are useful statistical inference procedures. Another type of inference is used frequently used concerns tests of hypotheses.
University of California, Los Angeles Department of Statistics Statistics 13 Elements of a hypothesis test: Hypothesis testing Instructor: Nicolas Christou 1. Null hypothesis, H 0 (always =). 2. Alternative
Review for Test 5 STA 2023 spr 2014 Name Given the linear correlation coefficient r and the sample size n, determine the critical values of r and use your finding to state whether or not the given r represents