# Unit 31: One-Way ANOVA

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 Unit 31: One-Way ANOVA Summary of Video A vase filled with coins takes center stage as the video begins. Students will be taking part in an experiment organized by psychology professor John Kelly in which they will guess the amount of money in the vase. As a subterfuge for the real purpose of the experiment, students are told that they are taking part in a study to test the theory of the Wisdom of the Crowd, which is that the average of all of the guesses will probably be more accurate than most of the individual guesses. However, the real purpose of the study is to see whether holding heavier or lighter clipboards while estimating the amount of money in the jar will have an impact on students guesses. The idea being tested is that physical experience can influence our thinking in ways we are unaware of this phenomenon is called embodied cognition. The sheet on which students will record their monetary guesses is clipped onto a clipboard. For the actual experiment, clipboards, each holding varying amounts of paper, weigh either one pound, two pounds or three pounds. Students are randomly assigned to clipboards and are unaware of any difference in the clipboards. After the data are collected, guesses are entered into a computer program and grouped according to the weights of the clipboards. The mean guess for each group is computed and the output is shown in Table Money Guesses Clipboard Weight Mean N Standard Deviation 1 \$ \$ \$ \$ \$ \$ Total \$ \$ Table 31.1 Table Average guesses by clipboard weight. Looking at the means, the results appear very promising. As clipboard weight goes up, so does the mean of the guesses, and that pattern appears fairly linear (See Figure 31.1.). To test whether or not the apparent differences in means could be due simply to chance, John turns to a technique called a one-way analysis of variance, or ANOVA. The null Figure Mean guess versus clipboard weight. Unit 31: One-Way ANOVA Student Guide Page 1

2 hypothesis for the analysis of variance will be that there is no difference in population means for the three weights of clipboards: H 0 : µ 1 = µ 2 = µ 3. He hopes to find sufficient evidence to reject the null hypothesis so that he can conclude that there is a significant difference among the population means. John runs an ANOVA using SPSS statistical software to compute a statistic called F, which is the ratio of two measures of variation: variation among sample means F= variation within individual observations in the same sample In this case, F = with a p-value of That means there is a 45% chance of getting an F value at least this extreme when there is no difference between the population means. So, the data from this experiment do not provide sufficient evidence to reject the null hypothesis. One of the underlying assumptions of ANOVA is that the data in each group are normally distributed. However, the boxplots in Figure 31.2 indicate that the data are skewed and include some rather extreme outliers. John s students tried some statistical manipulations on the data to make them more normal and reran the ANOVA. However, the conclusion remained the same. \$1, \$1, \$1, MoneyGuess \$1, \$ \$ \$ \$ \$ Clipboard Weight 3 Figure Boxplots of guesses grouped by clipboard weight. But what if we used the data displayed in Figure 31.3 instead? The sample means are the same, around \$107, \$130, and \$143, but this time the data are less spread out about those means. Unit 31: One-Way ANOVA Student Guide Page 2

3 MoneyGuess Clipboard Weight 3 Figure Hypothetical guess data. In this case, after running ANOVA, the result is F = with a p-value that is essentially zero. Our conclusion is to reject the null hypothesis and conclude that the population means are significantly different. In John s experiment, the harsh reality of a rigorous statistical analysis has shot down the idea that holding something heavy causes people, unconsciously, to make larger estimates, at least in this particular study. But if the real experiment didn t work, what about the cover story the theory of the Wisdom of the Crowd? The actual amount in the vase is \$ Figure 31.4 shows a histogram of all the guesses. The mean of the estimates is \$ more than \$100 off, but still better than about three-quarters of the individual guesses. So, the crowd was wiser than the people in it. Figure Histogram of guess data. Unit 31: One-Way ANOVA Student Guide Page 3

4 Student Learning Objectives A. Be able to identify when analysis of variance (ANOVA) should be used and what the null and alternative (research) hypotheses are. B. Be able to identify the factor(s) and response variable from a description of an experiment. C. Understand the basic logic of an ANOVA. Be able to describe between-sample variability (measured by mean square for groups (MSG)) and within-sample variability (measured by mean square error (MSE)). D. Know how to compute the F statistic and determine its degrees of freedom given the following summary statistics: sample sizes, sample means and sample standard deviations. Be able to use technology to compute the p-value for F. E. Be able to use technology to produce an ANOVA table. F. Recognize that statistically significant differences among population means depend on the size of the differences among the sample means, the amount of variation within the samples, and the sample sizes. G. Recognize when underlying assumptions for ANOVA are reasonably met so that it is appropriate to run an ANOVA. H. Be able to create appropriate graphic displays to support conclusions drawn from ANOVA output. Unit 31: One-Way ANOVA Student Guide Page 4

5 Content Overview In Unit 27, Comparing Two Means, we compared two population means, the mean total energy expenditure for Hadza and Western women, and used a two-sample t-test to test whether or not the means were equal. But what if you wanted to compare three population means? In that case, you could use a statistical procedure called Analysis of Variance or ANOVA, which was developed by Ronald Fisher in For example, suppose a statistics class wanted to test whether or not the amount of caffeine consumed affected memory. The variable caffeine is called a factor and students wanted to study how three levels of that factor affected the response variable, memory. Twelve students were recruited to take part in the study. The participants were divided into three groups of four and randomly assigned to one of the following drinks: A. Coca-Cola Classic (34 mg caffeine) B. McDonald s coffee (100 mg caffeine) C. Jolt Energy (160 mg caffeine). After drinking the caffeinated beverage, the participants were given a memory test (words remembered from a list). The results are given in Table Group A (34 mg) Group B (100 mg) Group C (160 mg) Table Number of words recalled in memory test. For an ANOVA, the null hypothesis is that the population means among the groups are the same. In this case, H 0 : µ A = µ B = µ C, where µ A is the population mean number of words recalled after people drink Coca Cola and similarly for µ B and µ C. The alternative or research hypothesis is that there is some inequality among the three means. Notice that there is a lot of variation in the number of words remembered by the participants. We break that variation into two components: (1) variation in the number of words recalled among the three groups also called between-groups variation Unit 31: One-Way ANOVA Student Guide Page 5

6 (2) variation in number of words among participants within each group also called within-groups variation. To measure each of these components, we ll compute two different variances, the mean square for groups (MSG) and the mean square error (MSE). The basic idea in gathering evidence to reject the null hypothesis is to show that the between-groups variation is substantially larger than the within-groups variation and we do that by forming the ratio, which we call F: between-groups variation F= within-groups variation =MSG MSE In the caffeine example, we have three groups. More generally, suppose there were k different groups (each assigned to consume varying amounts of caffeine) with sample sizes n 1, n 2, n k. Then the null hypothesis is H 0 : µ 1 = µ 2 =... = µ k and the alternative hypothesis is that at least two of the population means differ. The formulas for computing the between-groups variation and within-groups variation are given below: MSG = n 1 (x 1 - x)2 + n 2 (x 2 - x) n k (x k - x) 2 k -1 where x is the mean of all the observations and x 1,x 2,...,x k sample means for each group. are the MSE = (n -1)s 2 + (n )s ,+ (n k -1)s k N - k where N = n1+ n nk and s1, s2,..., s k are the sample standard deviations for each group. When H 0 is true, then F = MSG/MSE has the F distribution k 1 and N k degrees of freedom. We use the F distribution to calculate the p-value for the F-test statistic. We return to our three-group caffeine experiment to see how this works. To begin, we calculate the sample means and standard deviations (See Table 31.3.). Group Sample Mean Sample Standard Deviation A B C Table Table Group means and standard deviations. Unit 31: One-Way ANOVA Student Guide Page 6

7 Before calculating the mean square for groups we also need the grand mean of all the data values: x Now we are ready to calculate MSG, MSE, and F: MSG = 5( )2 + 5( ) 2 + 5( ) = = MSE = (5 1)(2.168)2 + (5 1)(1.789) 2 + (5 1)(2.236) = F = All that is left is to find the p-value. If the null hypothesis is true, then the F-statistic has the F distribution with 2 and 12 degrees of freedom. We use software to see how likely it would be to get an F value at least as extreme as Figure 31.5 shows the result giving a p-value of around Since p < 0.05, we conclude that the amount of caffeine consumed affected the mean memory score. Distribution Plot F, df1=2, df2= Density F 5.78 Figure Finding the p-value from the F-distribution. It takes a lot of work to compute F and find the p-value. Here s where technology can help. Statistical software such as Minitab, spreadsheet software such as Excel, and even graphing calculators can calculate ANOVA tables. Table 31.4 shows output from Minitab. Now, match the calculations above with the values in Table Check out where you can find the values for MSG, MSE, F, the degrees of freedom for F, and the p-value directly from the output of ANOVA. That will be a time saver! Unit 31: One-Way ANOVA Student Guide Page 7

8 Minitab. Now, match the calculations above with the values in Table Check out where you can find the values for,,, the degrees of freedom for, and the value directly from the output of ANOVA. That will be a time saver! Table ANOVA output from Minitab. Table ANOVA output from Minitab. It is important to understand that ANOVA does not tell you which population It means is important differ, to only understand that at least that two ANOVA of the does means not differ. tell you We which would population have to use means other differ, only tests that to help at least us decide two of the which means of the differ. three We population would have means to use are other significantly tests to different help us decide which from each of the other. three However, population we means can also are get significantly a clue by different plotting the from data. each Figure other However, shows comparative dotplots for the number of words for each group. The sample means we can also get a clue by plotting the data. Figure 31.6 shows comparative dotplots for the are marked with triangles. Notice that the biggest difference in sample means is number between of groups words for A (34 each mg group. caffeine) The and sample C (160 means mg of are caffeine). marked The with sample triangles. means Notice for that the groups biggest B and difference C are quite in sample close together. means is So, between it looks groups as if consuming A (34 mg caffeine) Coca Cola and doesn t C (160 mg give the memory boost you could expect from consuming coffee or Jolt Energy. of caffeine). The sample means for groups B and C are quite close together. So, it looks as if consuming Coca Cola doesn t give the memory boost you could expect from consuming coffee or Jolt Energy. Group A B C Figure Comparative dotplots Number of Words There is one last detail before jumping into running an ANOVA there are some Figure underlying assumptions Comparative that dotplots. need to be checked in order for the results of the analysis to be valid. What we should have done first with our caffeine experiment, we will do last. There Here is are one the last three detail things before to check. jumping into running an ANOVA there are some underlying assumptions that need to be checked in order for the results of the analysis to be valid. What 1. Each group s data need to be an independent random sample from that we should population. have done In first the case with our of an caffeine experiment, experiment, the subjects we will need do last. to be Here randomly are the three things to assigned check. to the levels of the factor. The subjects in the caffeine memory experiment were divided into 1. Each groups. group s Groups data need were to then be an randomly independent assigned random to the sample level of from caffeine. that population. In the case of an experiment, the subjects need to be randomly assigned to the levels of the Student factor. Guide, Unit 31, One Way ANOVA Page 7 Check: The subjects in the caffeine-memory experiment were divided into groups. Groups were then randomly assigned to the level of caffeine. Unit 31: One-Way ANOVA Student Guide Page 8

9 2. Next, each population has a Normal distribution. The results from ANOVA will be approximately correct as long as the sample group data are roughly normal. Problems can arise if the data are highly skewed or there are extreme outliers. Check: The normal quantile plots of Words Recalled for each group are shown in Figure Based on these plots, it seems reasonable to assume these data are from a Normal distribution. Normal Quantile Plots of Number of Words Normal - 95% CI 99 Group A (35 mg) 99 Group B (100 mg) Percent Group C (160 mg) Figure Checking the normality assumption. 3. All populations have the same standard deviation. The results from ANOVA will be approximately correct as long as the ratio of the largest standard deviation to the smallest standard deviation is less than 2. Check: The ratio of the largest to the smallest standard deviation is 2.236/1.789 or around 1.25, which is less than 2. Unit 31: One-Way ANOVA Student Guide Page 9

10 Key Terms A factor is a variable that can be used to differentiate population groups. The levels of a factor are the possible values or settings that a factor can assume. The variable of interest is the response variable, which may be related to one or more factors. An analysis of variance or ANOVA is a method of inference used to test whether or not three or more population means are equal. In a one-way ANOVA there is one factor that is thought to be related to the response variable. An analysis of variances tests the equality of means by comparing two types of variation, between-groups variation and within-groups variation. Between-groups variation deals with the spread of the group sample means about the grand mean, the mean of all the observations. It is measured by the mean square for groups, MSG. Within-groups variation deals with the spread of individual data values within a group about the group mean. It is measured by the mean square error, MSE. The F-statistic is the ratio MSG/MSE. Unit 31: One-Way ANOVA Student Guide Page 10

11 The Video Take out a piece of paper and be ready to write down answers to these questions as you watch the video. 1. What is the theory called the Wisdom of the Crowd? 2. What was different about the clipboards that students were holding? 3. State the null hypothesis for the one-way ANOVA. 4. What is the name of the test statistic that results from ANOVA? 5. Was the professor able to conclude from the F-statistic that the population means differed depending on the weight of the clipboard? Explain. 6. Did the crowd prove to be wiser than the individual students? Unit 31: One-Way ANOVA Student Guide Page 11

13 b. Compute the standard deviations for the three samples. Is the underlying assumption of equal standard deviations reasonably satisfied? Explain. c. Provided you answered yes to (b), use technology to run an ANOVA. State the null hypothesis being tested, the value of F, the p-value, and your conclusion. (If you answered no to (b), skip this part.) 3. Your final task will be to perform an experiment to collect data and determine whether settings for Control 3 affect the mean thickness of polished wafers. a. Leave Controls 1 and 2 set at level 2. Adapt the process used in question 1(a c) to collect the data on Control 3. b. Compute the standard deviations for the three samples. Is the underlying assumption of equal standard deviations reasonably satisfied? Explain. c. Provided you answered yes to (b), use technology to run an ANOVA. State the null hypothesis being tested, the value of F, the p-value, and your conclusion. (If you answered no to (b), skip this part.) Unit 31: One-Way ANOVA Student Guide Page 13

14 Exercises 1. A professor predicts that students will learn better if they study to white noise (similar to a fan) compared to music or no sound. She randomly divides 27 students into three groups and sends them to three different rooms. In the first room, students hear white noise, in the second, music from a local radio station, and in the third, the door is closed to help block out normal sound from the hall. Each group is given 30 minutes to study a section of their text after which they take a 10-question multiple-choice exam. Table 31.5 contains the results. White Noise Music No Sound Table Table Test results. a. Calculate the mean test score for each group. Calculate the standard deviation of the test scores for each group. b. Make comparative dotplots for the test results of the three groups. Do you think that the dotplots give sufficient evidence that there is a difference in population mean test results depending on the type of noise? Explain. c. Run an ANOVA. State the hypotheses you are testing. Show the calculations for the F-statistic. What are the degrees of freedom associated with this F-statistic? d. What is the p-value of the test statistic? What is your conclusion? 2. Not all hotdogs have the same calories. Table 31.6 contains calorie data on a random sample of Beef, Poultry, and Veggie dogs. (One extreme outlier for Veggie dogs was omitted from the data.) Does the mean calorie count differ depending on the type of hotdog? You first encountered this topic in Unit 5, Boxplots. Unit 31: One-Way ANOVA Student Guide Page 14

15 Calories Beef Poultry Veggie Table Calorie content of hotdogs. First-Year Cumulative Grade Point Average High Rating Medium Rating Low Rating Table Table First-year college GPA by high school rating. a. Verify that the standard deviations allow the use of ANOVA to compare population means. b. Use technology to run an ANOVA. State the value of the F-statistic, the degrees of freedom for F, the p-value of the test, and your conclusion. c. Make boxplots that compare the calorie data for each type of hot dog. Add a dot to each boxplot to mark the sample means. Do your plots help confirm your conclusion in (b)? 3. Many states rate their high schools using factors such as students performance, teachers educational backgrounds, and socioeconomic conditions. High school ratings for one state have been boiled down into three categories: high, medium, and low. The question for one of the state universities is whether or not college grade performance differs depending on high school rating. Table 31.7 contains random samples of students from each high school rating level and their first-year cumulative college grade point averages (GPA). a. Calculate the sample means for the GPAs in each group. Based on the sample means alone, does high school rating appear to have an impact on mean college GPA? Explain. b. Check to see that underlying assumptions for ANOVA are reasonably satisfied. Unit 31: One-Way ANOVA Student Guide Page 15

16 c. Run an ANOVA to test whether there is a significant difference among the population mean GPAs for the three groups. Report the value of the F-statistic, the p-value, and your conclusion. 4. Researchers in a nursing school of a large university conducted a study to determine if differences exist in levels of active and collaborative learning (ACL) among nursing students, other health professional students, and students majoring in education. A random sample of 1,000 students from each of these three majors was selected from students who completed the National Survey of Student Engagement (NSSE). a. The sample mean ACL scores for nursing, other health professional students, and education majors were 46.44, 45.58, and 48.59, respectively. Do these sample means provide sufficient evidence to conclude that there was some difference in population mean ACL scores among these three majors? Explain. b. A one-way analysis of variance was run to determine if there was a difference among the three groups on mean ACL scores. Assuming that all students answered the NSSE questions related to ACL, what were the degrees of freedom of the F-test? c. The results from the ANOVA gave F = Determine the p-value. What can you conclude? Unit 31: One-Way ANOVA Student Guide Page 16

17 Review Questions 1. Random samples of three types of candy were given to children. The three types of candy were chocolate bars (A), hard candy (B), and chewy candy (C). A group of 15 children rated samples of each type of candy on a scale from 1 (lowest) to 10 (highest). Two hypothetical data sets are given in Tables 31.8 and Ratings for A Ratings for B Ratings for C Table Data set #1. Table Data set #2. Table 31.8 Ratings for A Ratings for B Ratings for C Table a. Find the sample means of each candy type based on the ratings in Table Then do the same for the ratings in Table Based on these results, can you tell if there is a significant difference in population mean ratings among the different types of candies? Explain. b. Make comparative boxplots for the data in Table Then do the same for the data in Table For both sets of plots, mark the mean with a dot on each boxplot. For which data set is it more likely that the results from a one-way ANOVA will be significant? Explain. c. Run an ANOVA based on Data Set #1. Report the value of the F-statistic, the p-value, and your conclusion. Then do the same for Data Set #2. Explain why you should not be surprised by the results. 2. The data in Table were part of a study to investigate online questionnaire design. The researcher was interested in the effect that type of answer entry and type of question-toquestion navigation would have on the time it would take to complete online surveys. Twenty- Unit 31: One-Way ANOVA Student Guide Page 17

18 seven volunteers participated in this study. Each completed two questionnaires, one dealing with credit and the other focusing on vacations. Each questionnaire had 14 questions, and participants could select only one answer. There were three display types for answers: (1) radio button (2) drop down list (3) list box There were three navigational methods: (1) questions were on a single page (2) click the Next/Prev button (3) press Tab Display Type Navigation Type Time (sec) Display Type Navigation Type Time (sec) Table Time to complete on-line questionnaires. Unit 31: One-Way ANOVA Student Guide Page 18

20 a. Is the homogeneous standard deviations assumption for ANOVA reasonably satisfied? Explain. b. State the researchers null hypothesis. c. Calculate the value of the F-statistic and give its degrees of freedom. Show calculations. d. Determine the p-value. e. Based on the evidence in Table and your answers to (a d), what conclusions can the researchers make? 4. A study focusing on women s wages was investigating whether there was a significant difference in salaries in four occupations commonly (but not exclusively) held by women cashier, customer service representative, receptionist, and secretary/administrative assistant. Weekly wages from 50 women working in each occupation are recorded in Table Cashier Customer Service Receptionist Secretary/Adm. Asst Table Weekly wages of women in four occupations. Data from 2012 March Supplement, Current Population Survey. Unit 31: One-Way ANOVA Student Guide Page 20

21 a. Calculate the means and standard deviations for the weekly wages in each occupation category. b. Is the homogeneous standard deviations assumption for ANOVA reasonably satisfied? Explain. c. Run an ANOVA. Record the ANOVA table and highlight the value of F, and the p-value. d. Based on your answers to (a c), what are your conclusions? Unit 31: One-Way ANOVA Student Guide Page 21

### 1.5 Oneway Analysis of Variance

Statistics: Rosie Cornish. 200. 1.5 Oneway Analysis of Variance 1 Introduction Oneway analysis of variance (ANOVA) is used to compare several means. This method is often used in scientific or medical experiments

### 12: Analysis of Variance. Introduction

1: Analysis of Variance Introduction EDA Hypothesis Test Introduction In Chapter 8 and again in Chapter 11 we compared means from two independent groups. In this chapter we extend the procedure to consider

### Chapter 7. One-way ANOVA

Chapter 7 One-way ANOVA One-way ANOVA examines equality of population means for a quantitative outcome and a single categorical explanatory variable with any number of levels. The t-test of Chapter 6 looks

### ANOVA Analysis of Variance

ANOVA Analysis of Variance What is ANOVA and why do we use it? Can test hypotheses about mean differences between more than 2 samples. Can also make inferences about the effects of several different IVs,

### Unit 26: Small Sample Inference for One Mean

Unit 26: Small Sample Inference for One Mean Prerequisites Students need the background on confidence intervals and significance tests covered in Units 24 and 25. Additional Topic Coverage Additional coverage

### 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

### 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

### MINITAB ASSISTANT WHITE PAPER

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

### AP Statistics 2001 Solutions and Scoring Guidelines

AP Statistics 2001 Solutions and Scoring Guidelines The materials included in these files are intended for non-commercial use by AP teachers for course and exam preparation; permission for any other use

### Recall this chart that showed how most of our course would be organized:

Chapter 4 One-Way ANOVA Recall this chart that showed how most of our course would be organized: Explanatory Variable(s) Response Variable Methods Categorical Categorical Contingency Tables Categorical

### Section 9.3B Inference for Means: Paired Data

Section 9.3B Inference for Means: Paired Data 1 Objective PERFORM significance tests for paired data are called: paired t procedures. Comparative studies (i.e. 2 observations on 1 individual or 1 observation

### One-Way Analysis of Variance (ANOVA) Example Problem

One-Way Analysis of Variance (ANOVA) Example Problem Introduction Analysis of Variance (ANOVA) is a hypothesis-testing technique used to test the equality of two or more population (or treatment) means

### Simple Linear Regression Inference

Simple Linear Regression Inference 1 Inference requirements The Normality assumption of the stochastic term e is needed for inference even if it is not a OLS requirement. Therefore we have: Interpretation

### 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

### Experimental Designs (revisited)

Introduction to ANOVA Copyright 2000, 2011, J. Toby Mordkoff Probably, the best way to start thinking about ANOVA is in terms of factors with levels. (I say this because this is how they are described

### ANOVA ANOVA. Two-Way ANOVA. One-Way ANOVA. When to use ANOVA ANOVA. Analysis of Variance. Chapter 16. A procedure for comparing more than two groups

ANOVA ANOVA Analysis of Variance Chapter 6 A procedure for comparing more than two groups independent variable: smoking status non-smoking one pack a day > two packs a day dependent variable: number of

### 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

### Chapter 7 Section 7.1: Inference for the Mean of a Population

Chapter 7 Section 7.1: Inference for the Mean of a Population Now let s look at a similar situation Take an SRS of size n Normal Population : N(, ). Both and are unknown parameters. Unlike what we used

### An example ANOVA situation. 1-Way ANOVA. Some notation for ANOVA. Are these differences significant? Example (Treating Blisters)

An example ANOVA situation Example (Treating Blisters) 1-Way ANOVA MATH 143 Department of Mathematics and Statistics Calvin College Subjects: 25 patients with blisters Treatments: Treatment A, Treatment

### Statistics courses often teach the two-sample t-test, linear regression, and analysis of variance

2 Making Connections: The Two-Sample t-test, Regression, and ANOVA In theory, there s no difference between theory and practice. In practice, there is. Yogi Berra 1 Statistics courses often teach the two-sample

### Statistiek II. John Nerbonne. October 1, 2010. Dept of Information Science j.nerbonne@rug.nl

Dept of Information Science j.nerbonne@rug.nl October 1, 2010 Course outline 1 One-way ANOVA. 2 Factorial ANOVA. 3 Repeated measures ANOVA. 4 Correlation and regression. 5 Multiple regression. 6 Logistic

### Statistics Review PSY379

Statistics Review PSY379 Basic concepts Measurement scales Populations vs. samples Continuous vs. discrete variable Independent vs. dependent variable Descriptive vs. inferential stats Common analyses

### DEPARTMENT OF PSYCHOLOGY UNIVERSITY OF LANCASTER MSC IN PSYCHOLOGICAL RESEARCH METHODS ANALYSING AND INTERPRETING DATA 2 PART 1 WEEK 9

DEPARTMENT OF PSYCHOLOGY UNIVERSITY OF LANCASTER MSC IN PSYCHOLOGICAL RESEARCH METHODS ANALYSING AND INTERPRETING DATA 2 PART 1 WEEK 9 Analysis of covariance and multiple regression So far in this course,

### Testing Group Differences using T-tests, ANOVA, and Nonparametric Measures

Testing Group Differences using T-tests, ANOVA, and Nonparametric Measures Jamie DeCoster Department of Psychology University of Alabama 348 Gordon Palmer Hall Box 870348 Tuscaloosa, AL 35487-0348 Phone:

### c. The factor is the type of TV program that was watched. The treatment is the embedded commercials in the TV programs.

STAT E-150 - Statistical Methods Assignment 9 Solutions Exercises 12.8, 12.13, 12.75 For each test: Include appropriate graphs to see that the conditions are met. Use Tukey's Honestly Significant Difference

### Business Statistics. Successful completion of Introductory and/or Intermediate Algebra courses is recommended before taking Business Statistics.

Business Course Text Bowerman, Bruce L., Richard T. O'Connell, J. B. Orris, and Dawn C. Porter. Essentials of Business, 2nd edition, McGraw-Hill/Irwin, 2008, ISBN: 978-0-07-331988-9. Required Computing

### Analysis of Variance. MINITAB User s Guide 2 3-1

3 Analysis of Variance Analysis of Variance Overview, 3-2 One-Way Analysis of Variance, 3-5 Two-Way Analysis of Variance, 3-11 Analysis of Means, 3-13 Overview of Balanced ANOVA and GLM, 3-18 Balanced

### Unit 21 Student s t Distribution in Hypotheses Testing

Unit 21 Student s t Distribution in Hypotheses Testing Objectives: To understand the difference between the standard normal distribution and the Student's t distributions To understand the difference between

EC151.02 Statistics for Business and Economics (MWF 8:00-8:50) Instructor: Chiu Yu Ko Office: 462D, 21 Campenalla Way Phone: 2-6093 Email: kocb@bc.edu Office Hours: by appointment Description This course

### 1.3 Measuring Center & Spread, The Five Number Summary & Boxplots. Describing Quantitative Data with Numbers

1.3 Measuring Center & Spread, The Five Number Summary & Boxplots Describing Quantitative Data with Numbers 1.3 I can n Calculate and interpret measures of center (mean, median) in context. n Calculate

### ANOVA MULTIPLE CHOICE QUESTIONS. In the following multiple-choice questions, select the best answer.

ANOVA MULTIPLE CHOICE QUESTIONS In the following multiple-choice questions, select the best answer. 1. Analysis of variance is a statistical method of comparing the of several populations. a. standard

### 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

### INTERPRETING THE ONE-WAY ANALYSIS OF VARIANCE (ANOVA)

INTERPRETING THE ONE-WAY ANALYSIS OF VARIANCE (ANOVA) As with other parametric statistics, we begin the one-way ANOVA with a test of the underlying assumptions. Our first assumption is the assumption of

### Chapter 9. Two-Sample Tests. Effect Sizes and Power Paired t Test Calculation

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

### Main Effects and Interactions

Main Effects & Interactions page 1 Main Effects and Interactions So far, we ve talked about studies in which there is just one independent variable, such as violence of television program. You might randomly

### Course Text. Required Computing Software. Course Description. Course Objectives. StraighterLine. Business Statistics

Course Text Business Statistics Lind, Douglas A., Marchal, William A. and Samuel A. Wathen. Basic Statistics for Business and Economics, 7th edition, McGraw-Hill/Irwin, 2010, ISBN: 9780077384470 [This

### SAS / INSIGHT. ShortCourse Handout

SAS / INSIGHT ShortCourse Handout February 2005 Copyright 2005 Heide Mansouri, Technology Support, Texas Tech University. ALL RIGHTS RESERVED. Members of Texas Tech University or Texas Tech Health Sciences

### Business Statistics. Lecture 8: More Hypothesis Testing

Business Statistics Lecture 8: More Hypothesis Testing 1 Goals for this Lecture Review of t-tests Additional hypothesis tests Two-sample tests Paired tests 2 The Basic Idea of Hypothesis Testing Start

### find confidence interval for a population mean when the population standard deviation is KNOWN Understand the new distribution the t-distribution

Section 8.3 1 Estimating a Population Mean Topics find confidence interval for a population mean when the population standard deviation is KNOWN find confidence interval for a population mean when the

### One-Way Analysis of Variance

One-Way Analysis of Variance Note: Much of the math here is tedious but straightforward. We ll skim over it in class but you should be sure to ask questions if you don t understand it. I. Overview A. We

### 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

### SPSS Manual for Introductory Applied Statistics: A Variable Approach

SPSS Manual for Introductory Applied Statistics: A Variable Approach John Gabrosek Department of Statistics Grand Valley State University Allendale, MI USA August 2013 2 Copyright 2013 John Gabrosek. All

### 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

### Data Analysis Tools. Tools for Summarizing Data

Data Analysis Tools This section of the notes is meant to introduce you to many of the tools that are provided by Excel under the Tools/Data Analysis menu item. If your computer does not have that tool

### Technology Step-by-Step Using StatCrunch

Technology Step-by-Step Using StatCrunch Section 1.3 Simple Random Sampling 1. Select Data, highlight Simulate Data, then highlight Discrete Uniform. 2. Fill in the following window with the appropriate

### Chapter 6: t test for dependent samples

Chapter 6: t test for dependent samples ****This chapter corresponds to chapter 11 of your book ( t(ea) for Two (Again) ). What it is: The t test for dependent samples is used to determine whether the

### Statistical Inference and t-tests

1 Statistical Inference and t-tests Objectives Evaluate the difference between a sample mean and a target value using a one-sample t-test. Evaluate the difference between a sample mean and a target value

### 1.5 NUMERICAL REPRESENTATION OF DATA (Sample Statistics)

1.5 NUMERICAL REPRESENTATION OF DATA (Sample Statistics) As well as displaying data graphically we will often wish to summarise it numerically particularly if we wish to compare two or more data sets.

### Unit 29 Chi-Square Goodness-of-Fit Test

Unit 29 Chi-Square Goodness-of-Fit Test Objectives: To perform the chi-square hypothesis test concerning proportions corresponding to more than two categories of a qualitative variable To perform the Bonferroni

### Box plots & t-tests. Example

Box plots & t-tests Box Plots Box plots are a graphical representation of your sample (easy to visualize descriptive statistics); they are also known as box-and-whisker diagrams. Any data that you can

### KSTAT MINI-MANUAL. Decision Sciences 434 Kellogg Graduate School of Management

KSTAT MINI-MANUAL Decision Sciences 434 Kellogg Graduate School of Management Kstat is a set of macros added to Excel and it will enable you to do the statistics required for this course very easily. To

### Hypothesis Testing Level I Quantitative Methods. IFT Notes for the CFA exam

Hypothesis Testing 2014 Level I Quantitative Methods IFT Notes for the CFA exam Contents 1. Introduction... 3 2. Hypothesis Testing... 3 3. Hypothesis Tests Concerning the Mean... 10 4. Hypothesis Tests

### Hypothesis Testing & Data Analysis. Statistics. Descriptive Statistics. What is the difference between descriptive and inferential statistics?

2 Hypothesis Testing & Data Analysis 5 What is the difference between descriptive and inferential statistics? Statistics 8 Tools to help us understand our data. Makes a complicated mess simple to understand.

### AP Statistics 2008 Scoring Guidelines Form B

AP Statistics 2008 Scoring Guidelines Form B The College Board: Connecting Students to College Success The College Board is a not-for-profit membership association whose mission is to connect students

### One-sample normal hypothesis Testing, paired t-test, two-sample normal inference, normal probability plots

1 / 27 One-sample normal hypothesis Testing, paired t-test, two-sample normal inference, normal probability plots Timothy Hanson Department of Statistics, University of South Carolina Stat 704: Data Analysis

### Final Exam Practice Problem Answers

Final Exam Practice Problem Answers The following data set consists of data gathered from 77 popular breakfast cereals. The variables in the data set are as follows: Brand: The brand name of the cereal

### Chapter 7: Simple linear regression Learning Objectives

Chapter 7: Simple linear regression Learning Objectives Reading: Section 7.1 of OpenIntro Statistics Video: Correlation vs. causation, YouTube (2:19) Video: Intro to Linear Regression, YouTube (5:18) -

### Chapter 5 Analysis of variance SPSS Analysis of variance

Chapter 5 Analysis of variance SPSS Analysis of variance Data file used: gss.sav How to get there: Analyze Compare Means One-way ANOVA To test the null hypothesis that several population means are equal,

### STATS8: Introduction to Biostatistics. Data Exploration. Babak Shahbaba Department of Statistics, UCI

STATS8: Introduction to Biostatistics Data Exploration Babak Shahbaba Department of Statistics, UCI Introduction After clearly defining the scientific problem, selecting a set of representative members

### CHAPTER 14 NONPARAMETRIC TESTS

CHAPTER 14 NONPARAMETRIC TESTS Everything that we have done up until now in statistics has relied heavily on one major fact: that our data is normally distributed. We have been able to make inferences

### Once saved, if the file was zipped you will need to unzip it. For the files that I will be posting you need to change the preferences.

1 Commands in JMP and Statcrunch Below are a set of commands in JMP and Statcrunch which facilitate a basic statistical analysis. The first part concerns commands in JMP, the second part is for analysis

### Hypothesis Tests Means 2 samples

Hypothesis Tests Means 2 samples 1. More eggs? Can a food additive increase egg production? Agricultural researchers want to design an experiment to find out. They have 100 hens available. They have two

### 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

### Provide an appropriate response. Solve the problem. Determine the null and alternative hypotheses for the proposed hypothesis test.

Provide an appropriate response. 1) Suppose that x is a normally distributed variable on each of two populations. Independent samples of sizes n1 and n2, respectively, are selected from the two populations.

### Two-sample t-tests. - Independent samples - Pooled standard devation - The equal variance assumption

Two-sample t-tests. - Independent samples - Pooled standard devation - The equal variance assumption Last time, we used the mean of one sample to test against the hypothesis that the true mean was a particular

### Research Variables. Measurement. Scales of Measurement. Chapter 4: Data & the Nature of Measurement

Chapter 4: Data & the Nature of Graziano, Raulin. Research Methods, a Process of Inquiry Presented by Dustin Adams Research Variables Variable Any characteristic that can take more than one form or value.

### NCSS Statistical Software Principal Components Regression. In ordinary least squares, the regression coefficients are estimated using the formula ( )

Chapter 340 Principal Components Regression Introduction is a technique for analyzing multiple regression data that suffer from multicollinearity. When multicollinearity occurs, least squares estimates

### 2011 # AP Exam Solutions 2011 # # # #1 5/16/2011

2011 AP Exam Solutions 1. A professional sports team evaluates potential players for a certain position based on two main characteristics, speed and strength. (a) Speed is measured by the time required

### Quantitative Data Analysis: Choosing a statistical test Prepared by the Office of Planning, Assessment, Research and Quality

Quantitative Data Analysis: Choosing a statistical test Prepared by the Office of Planning, Assessment, Research and Quality 1 To help choose which type of quantitative data analysis to use either before

### Center: Finding the Median. Median. Spread: Home on the Range. Center: Finding the Median (cont.)

Center: Finding the Median When we think of a typical value, we usually look for the center of the distribution. For a unimodal, symmetric distribution, it s easy to find the center it s just the center

### An analysis method for a quantitative outcome and two categorical explanatory variables.

Chapter 11 Two-Way ANOVA An analysis method for a quantitative outcome and two categorical explanatory variables. If an experiment has a quantitative outcome and two categorical explanatory variables that

### Lab 6: Sampling Distributions and the CLT

Lab 6: Sampling Distributions and the CLT Objective: The objective of this lab is to give you a hands- on discussion and understanding of sampling distributions and the Central Limit Theorem (CLT), a theorem

### MAT 12O ELEMENTARY STATISTICS I

LAGUARDIA COMMUNITY COLLEGE CITY UNIVERSITY OF NEW YORK DEPARTMENT OF MATHEMATICS, ENGINEERING, AND COMPUTER SCIENCE MAT 12O ELEMENTARY STATISTICS I 3 Lecture Hours, 1 Lab Hour, 3 Credits Pre-Requisite:

### 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

### Walk the Line Written by: Maryann Huey Drake University Maryann.Huey@drake.edu

Walk the Line Written by: Maryann Huey Drake University Maryann.Huey@drake.edu Overview of Lesson In this activity, students will conduct an investigation to collect data to determine how far students

### When to use Excel. When NOT to use Excel 9/24/2014

Analyzing Quantitative Assessment Data with Excel October 2, 2014 Jeremy Penn, Ph.D. Director When to use Excel You want to quickly summarize or analyze your assessment data You want to create basic visual

### Analysis of Student Retention Rates and Performance Among Fall, 2013 Alternate College Option (ACO) Students

1 Analysis of Student Retention Rates and Performance Among Fall, 2013 Alternate College Option (ACO) Students A Study Commissioned by the Persistence Committee Policy Working Group prepared by Jeffrey

### A Logic of Prediction and Evaluation

5 - Hypothesis Testing in the Linear Model Page 1 A Logic of Prediction and Evaluation 5:12 PM One goal of science: determine whether current ways of thinking about the world are adequate for predicting

### Solutions 7. Review, one sample t-test, independent two-sample t-test, binomial distribution, standard errors and one-sample proportions.

Solutions 7 Review, one sample t-test, independent two-sample t-test, binomial distribution, standard errors and one-sample proportions. (1) Here we debunk a popular misconception about confidence intervals

### 0.1 Multiple Regression Models

0.1 Multiple Regression Models We will introduce the multiple Regression model as a mean of relating one numerical response variable y to two or more independent (or predictor variables. We will see different

### The scatterplot indicates a positive linear relationship between waist size and body fat percentage:

STAT E-150 Statistical Methods Multiple Regression Three percent of a man's body is essential fat, which is necessary for a healthy body. However, too much body fat can be dangerous. For men between the

### Guide for SPSS for Windows

Guide for SPSS for Windows Index Table Open an existing data file Open a new data sheet Enter or change data value Name a variable Label variables and data values Enter a categorical data Delete a record

### Chapter 8 Hypothesis Testing

Chapter 8 Hypothesis Testing Chapter problem: Does the MicroSort method of gender selection increase the likelihood that a baby will be girl? MicroSort: a gender-selection method developed by Genetics

### , then the form of the model is given by: which comprises a deterministic component involving the three regression coefficients (

Multiple regression Introduction Multiple regression is a logical extension of the principles of simple linear regression to situations in which there are several predictor variables. For instance if we

### Inferential Statistics

Inferential Statistics Sampling and the normal distribution Z-scores Confidence levels and intervals Hypothesis testing Commonly used statistical methods Inferential Statistics Descriptive statistics are

### 7. Comparing Means Using t-tests.

7. Comparing Means Using t-tests. Objectives Calculate one sample t-tests Calculate paired samples t-tests Calculate independent samples t-tests Graphically represent mean differences In this chapter,

### AP Statistics 2010 Scoring Guidelines

AP Statistics 2010 Scoring Guidelines The College Board The College Board is a not-for-profit membership association whose mission is to connect students to college success and opportunity. Founded in

### Sociology 6Z03 Topic 15: Statistical Inference for Means

Sociology 6Z03 Topic 15: Statistical Inference for Means John Fox McMaster University Fall 2016 John Fox (McMaster University) Soc 6Z03: Statistical Inference for Means Fall 2016 1 / 41 Outline: Statistical

### An SPSS companion book. Basic Practice of Statistics

An SPSS companion book to Basic Practice of Statistics SPSS is owned by IBM. 6 th Edition. Basic Practice of Statistics 6 th Edition by David S. Moore, William I. Notz, Michael A. Flinger. Published by

### Chapter 1: Exploring Data

Chapter 1: Exploring Data Chapter 1 Review 1. As part of survey of college students a researcher is interested in the variable class standing. She records a 1 if the student is a freshman, a 2 if the student

### ABSORBENCY OF PAPER TOWELS

ABSORBENCY OF PAPER TOWELS 15. Brief Version of the Case Study 15.1 Problem Formulation 15.2 Selection of Factors 15.3 Obtaining Random Samples of Paper Towels 15.4 How will the Absorbency be measured?

### Difference of Means and ANOVA Problems

Difference of Means and Problems Dr. Tom Ilvento FREC 408 Accounting Firm Study An accounting firm specializes in auditing the financial records of large firm It is interested in evaluating its fee structure,particularly

### 7.1 Inference for comparing means of two populations

Objectives 7.1 Inference for comparing means of two populations Matched pair t confidence interval Matched pair t hypothesis test http://onlinestatbook.com/2/tests_of_means/correlated.html Overview of

### Chapter 11: Linear Regression - Inference in Regression Analysis - Part 2

Chapter 11: Linear Regression - Inference in Regression Analysis - Part 2 Note: Whether we calculate confidence intervals or perform hypothesis tests we need the distribution of the statistic we will use.

### Simple Linear Regression in SPSS STAT 314

Simple Linear Regression in SPSS STAT 314 1. Ten Corvettes between 1 and 6 years old were randomly selected from last year s sales records in Virginia Beach, Virginia. The following data were obtained,

### AP Statistics 2007 Scoring Guidelines

AP Statistics 2007 Scoring Guidelines The College Board: Connecting Students to College Success The College Board is a not-for-profit membership association whose mission is to connect students to college