MGMT 276: Statistical Inference in Management

MGMT 276: Statistical Inference in Management Welcome Please double check – All cell phones other electronic devices are turned off and stowed away http://www.thedailyshow.com/video/index.jhtml?videoId=188474&title=an-arab-family-man

Please read: Chapters 10 – 12 in Lind book and Chapters 2 – 4 in Plous book: (Before the next exam) Lind Chapter 10: One sample Tests of Hypothesis Chapter 11: Two sample Tests of Hypothesis Chapter 12: Analysis of Variance Plous Chapter 2: Cognitive Dissonance Chapter 3: Memory and Hindsight Bias Chapter 4: Context Dependence

Please click in Homework due next class - (Due April 5th) My last name starts with a letter somewhere between A. A – D B. E – L C. M – R D. S – Z Complete worksheet available on class website Please hand in your homework. Remember CID# Please double check – All cell phones other electronic devices are turned off and stowed away Turn your clicker on

Exam 3 – This Thursday (4/7/11) Study guide will go up tomorrow Bring 2 pencils (with good erasers) Bring ID Bring Calculator

Use this as your study guide By the end of lecture today3/31/11 Logic of hypothesis testing Steps for hypothesis testing Interpreting excel output of hypothesis tests Constructing brief, complete summary statements Hypothesis testing with analysis of variance (ANOVA) One-way versus Two-way ANOVA

Five steps to hypothesis testing Step 1: Identify the research problem (hypothesis) Describe the null and alternative hypotheses Step 2: Decision rule • Alpha level? (α= .05 or .01)? Still, difference between means • Critical statistic (e.g. z or t or F or r) value? Step 3: Calculations MSBetween F = MSWithin Still, variabilityof curve(s) Step 4: Make decision whether or not to reject null hypothesis If observed t (or F) is bigger then critical t (or F) then reject null Step 5: Conclusion - tie findings back in to research problem

“Between Groups”Differences . Difference between means Difference between means Difference between means Variabilityof curve(s) “Within Groups”Variability Variabilityof curve(s) Variabilityof curve(s)

One way analysis of varianceVariance is divided Remember, one-way = one IV Total variability Between group variability (only one factor) Within group variability (error variance) Remember, 1 factor = 1 independent variable(this will be our numerator – like difference between means) Remember, error variance = random error(this will be our denominator – within group variability)

One-way ANOVA Number of cookies sold Bike None Hawaii trip Incentives • One-way ANOVAs test only one independent variable • - although there may be many levels • “Factor” = one independent variable • “Level” = levels of the independent variable • treatment • condition • groups • “Main Effect” of independent variable = difference between levels • Note: doesn’t tell you which specific levels (means) differ from each other A multi-factor experiment would be a multi-independent variables experiment

Comparing ANOVAs with t-tests Similarities still include: Using distributions to make decisions about common and rare events Using distributions to make inferences about whether to reject the null hypothesis or not The same 5 steps for testing an hypothesis Tells us generally about number of participants / observations Tells us generally about number of groups / levels of IV • The three primary differences between t-tests and ANOVAS are: • 1. ANOVAs can test more than two means • 2. We are comparing sample means indirectly by • comparing sample variances • 3. We now will have two types of degrees of freedom • t(16) = 3.0; p < 0.05 F(2, 16) = 3.0; p < 0.05 Tells us generally about number of participants / observations

Three different types of variance Between groups Within groups Total Between Groups Variability Total Variability Variability between groups F = Within Groups Variability Variability within groups

ANOVA Variability between groups F = Variability within groups Variability Between Groups “Between” variability bigger than “within” variability so should get a big (significant) F Variability Within Groups Variability Within Groups Variability Between Groups “Between” variability getting smaller “within” variability staying same so, should get a smaller F Variability Within Groups Variability Within Groups Variability Between Groups “Between” variability getting very small “within” variability staying same so, should get a very small F Variability Within Groups Variability Within Groups

ANOVA Variability between groups F = Variability within groups Variability Between Groups “Between” variability bigger than “within” variability so should get a big (significant) F Variability Within Groups Variability Within Groups Variability Between Groups “Between” variability getting smaller “within” variability staying same so, should get a smaller F Variability Within Groups “Between” variability getting very small “within” variability staying same so, should get a very small F (equal to 1)

. Effect size is considered relativeto variability of distributions Treatment Effect x Variability between groups Treatment Effect x Variabilitywithin groups

x = 1 x = 6 x = 5 What if we want to compare 3 means? One independent variable with 3 means A girlscout troop leader wondered whether providing an incentive to whomever sold the most girlscout cookies would have an effect on the number cookies sold. She provided a big incentive to one troop (trip to Hawaii), a lesser incentive to a second troop (bicycle), and no incentive to a third group, and then looked to see who sold more cookies. Troop 1 (Hawaii) 6 5 9 4 6 Troop 2 (bicycle) 6 8 5 4 2 Troop 3 (nada) 0 4 0 1 0 Note: 5 girls in each troop

Main effect of incentive Number of cookies sold Number of cookies sold Hawaii None Bike None Bike Hawaii Incentives Incentives

A girl scout troop leader wondered whether providing an incentive to whomever sold the most girl scout cookies would have an effect on the number cookies sold. She provided a big incentive to one troop (trip to Hawaii), a lesser incentive to a second troop (bicycle), and no incentive to a third group, and then looked to see who sold more cookies. How many levels of the Independent Variable? What is Independent Variable? Troop 3 (Hawaii) 14 9 19 13 15 Troop 1 (nada) 10 8 12 7 13 Troop 2 (bicycle) 12 14 10 11 13 What is Dependent Variable? How many groups? n = 5 x = 10 n = 5 x = 12 n = 5 x = 14

Main effect of incentive: Will offering an incentive result in more girl scout cookies being sold? • If we have a “effect” of • incentive then the means • are significantly different • from each other • we reject the null • we have a significant F • p < 0.05 • To get an effect we want: • Large “F” - big effect and small variability • Small “p” - less than 0.05 (whatever our alpha is) To figure out which means are different from which …. we will do “post-hoc” t-tests We don’t know which means are different from which …. just that they are not all the same

Hypothesis testing: Step 1: Identify the research problem Is there a significant difference in the number of cookie boxes sold between the girlscout troops that were given the different levels of incentive? Describe the null and alternative hypotheses

Hypothesis testing: = .05 Decision rule Degrees of freedom (between) = number of groups - 1 = 3 - 1 = 2 Degrees of freedom (within) = # of scores - # of groups = (15-3) = 12* Critical F(2,12) = 3.98 *or = (5-1) + (5-1) + (5-1) = 12.

Appendix B.4 (pg.518) F (2,12) α= .05 Critical F(2,12) = 3.89

“SS” = “Sum of Squares”- will be given for exams- you can think of this as the numerator in a standard deviation formula ANOVA table F Source df MS SS Between ? ? ? ? Within ? ? ? Total ? ?

“SS” = “Sum of Squares”- will be given for exams ANOVA table F Source df MS SS 3-1=2 # groups - 1 Between 40 ? 2 ? ? ? 15-3=12 Within ? 88 ? 12 # scores - number of groups ? Total ? 128 ? 14 # scores - 1 15- 1=14

ANOVA table MSbetween MSwithin 40 88 SSbetween 12 2 ANOVA table dfbetween F Source df MS SS ? Between 40 2 ? 2.73 20 Within 88 12 ? 7.33 Total 128 14 SSwithin dfwithin 88 20 =2.73 =7.33 40 7.33 12 =20 2

Make decision whether or not to reject null hypothesis Observed F = 2.73 Critical F(2,12) = 3.89 2.73 is not farther out on the curve than 3.89 so, we do not reject the null hypothesis F(2,12) = 2.73; n.s. Conclusion: There appears to be no effect of type of incentive on number of girl scout cookies sold The average number of cookies sold for three different incentives were compared. The mean number of cookie boxes sold for the “Hawaii” incentive was 14 , the mean number of cookies boxes sold for the “Bicycle” incentive was 12, and the mean number of cookies sold for the “No” incentive was 10. An ANOVA was conducted and there appears to be no significant difference in the number of cookies sold as a result of the different levels of incentive F(2, 12) = 2.73; n.s.

Let’s do same problemUsing MS Excel A girlscout troop leader wondered whether providing an incentive to whomever sold the most girlscout cookies would have an effect on the number cookies sold. She provided a big incentive to one troop (trip to Hawaii), a lesser incentive to a second troop (bicycle), and no incentive to a third group, and then looked to see who sold more cookies. Troop 1 (Nada) 10 8 12 7 13 Troop 2 (bicycle) 12 14 10 11 13 Troop 3 (Hawaii) 14 9 19 13 15 n = 5 x = 10 n = 5 x = 12 n = 5 x = 14

Let’s do same problemUsing MS Excel

Let’s do oneReplication of study(new data)

Let’s do same problemUsing MS Excel

MSbetween MSwithin 40 88 SSbetween 12 2 dfbetween 3-1=2 # groups - 1 SSwithin dfwithin # scores - number of groups 15-3=12 88 20 =2.73 =7.33 40 # scores - 1 7.33 12 =20 2 15- 1=14

No, so it is not significant Do not reject null No, so it is not significant Do not reject null F critical(is observed F greater than critical F?) P-value(is it less than .05?)

Make decision whether or not to reject null hypothesis Observed F = 2.73 Critical F(2,12) = 3.89 2.7 is not farther out on the curve than 3.89 so, we do not reject the null hypothesis Also p-value is not smaller than 0.05 so we do not reject the null hypothesis Step 6: Conclusion: There appears to be no effect of type of incentive on number of girl scout cookies sold

Make decision whether or not to reject null hypothesis Observed F = 2.72727272 F(2,12) = 2.73; n.s. Critical F(2,12) = 3.88529 2.7 is not farther out on the curve than 3.89 so, we do not reject the null hypothesis Conclusion: There appears to be no effect of type of incentive on number of girl scout cookies sold The average number of cookies sold for three different incentives were compared. The mean number of cookie boxes sold for the “Hawaii” incentive was 14 , the mean number of cookies boxes sold for the “Bicycle” incentive was 12, and the mean number of cookies sold for the “No” incentive was 10. An ANOVA was conducted and there appears to be no significant difference in the number of cookies sold as a result of the different levels of incentive F(2, 12) = 2.73; n.s.

ANOVA - Analysis of VarianceThe larger the ratio (F) the bigger the effect(taking into account the error variance) • When the null hypothesis is true: • the ratio of the between-groups population variance (estimate) to • the within-groups population variance (estimate) should be about 1. • When the research hypothesis is true, • this ratio should be greater than 1 Variability between groups F = Variability within groups MSBetween F = MSWithin

When the null hypothesis is true: • the ratio of the between-groups population variance (estimate) to • the within-groups population variance (estimate) should be about 1. • the three means are equal • When the alternative hypothesis is true, • this ratio should be greater than 1 (F will be large & p will be small) By comparing the within group and between group variance we are able to answer questions about whether the differences between the means of groups are significant

Writing assignment worksheet:Propose an experiment you can actually do • Propose an experiment that would consist of one independent • variable (IV) and one dependent variable (DV). The IV should • have three groups (or more). The design should be appropriate • for an analysis that uses an ANOVA • What is your question / What is your prediction • What is your IV • How many levels does it have • What are the levels • What is your DV • How many subjects do you think you can gather data on? • Sketch a bar graph of your predicted results Keep a copy for yourself… You will need it

ANOVA Project - Due April 19th - There are five parts • 1. A one page report of your design (includes all of the information from the writing assignment) • Describe your experiment: what is your question / what is your prediction? • State your Independent Variable (IV), how many levels there are, and the operational definition • State your Dependent Variable (DV), and operational definition • How many participants did you measure, and how did you recruit (sample) them • Was this a between or within participant design (why?) • 2. Gather the data • Try to get at least 10 people (or data points) per level • If you are working with other students in the class you should have 10 data points per level for each member of your group • 3. Input data into Excel (hand in data) • 4. Complete ANOVA analysis hand in ANOVA table • 5. Statement of results (see next slide for example) and include • a graph of your means (just like we did in the homework)

How to report the findings for an ANOVA One paragraph summary of this study. Describe the IV & DV, and present the means, which type of test was conducted, and the statistical results. Start summary with two means (based on DV) for two levels of the IV The average number of cookies sold for three different incentives were compared. The mean number of cookie boxes sold for the “Hawaii” incentive was 14 , the mean number of cookies boxes sold for the “Bicycle” incentive was 12, and the mean number of cookies sold for the “No” incentive was 10. An ANOVA was conducted and there appears to be no significant difference in the number of cookies sold as a result of the different levels of incentive F(2, 12) = 2.73; n.s. Type of test with degrees of freedom Describe type of test (t-test versus anova) with brief overview of results Value of observed statistic p<0.05 = “significant”

Thank you! See you next time!!

MGMT 276: Statistical Inference in Management