Screen Stage Lecturer’s desk Gallagher Theater Row A Row A Row A Row B 17 16 15 14 13 12 11 10 9 8 7 6 5 4 Row A 3 2 1 Row A Left handed Row B 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 Row B 4 3 2 1 Row B Row C 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 Row C 4 3 2 1 Row C Row D 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 Row D 4 3 2 1 Row D Row E 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 Row E 4 3 2 1 Row E Row F 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 Row F 4 3 2 1 Row F Row G 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 Row G 4 3 2 1 Row G Row H 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 Row H 4 3 2 1 Row H Row I 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 Row I 4 3 2 1 Row I Row J 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 Row J 4 3 2 1 Row J Row K 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 Row K 4 3 2 1 Row K Row L 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 Row L 4 3 2 1 Row L Row M 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 Row M 4 3 2 1 Row M Row N 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 Row N 4 3 2 1 Row N Row O 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 Row O 4 3 2 1 Row O Need Labels B5, E1, I16, J17, K8, M4, O1, P16 Row P 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 Row P 4 3 2 1 Row P Row Q 16 15 14 13 12 11 10 9 8 7 6 5 4 Row Q 3 2 1 Row Q Row R Gallagher Theater 4 3 2 Row R 26Left-Handed Desks A14, B16, B20, C19, D16, D20, E15, E19, F16, F20, G19, H16, H20, I15, J16, J20, K19, L16, L20, M15, M19, N16, P20, Q13, Q16, S4 5 Broken Desks B9, E12, G9, H3, M17 Row S 10 9 8 7 4 3 2 1 Row S
Screen Stage Social Sciences 100 Lecturer’s desk broken desk R/L handed Row A 17 16 15 14 13 12 Row B 27 26 25 24 23 Row B 22 21 20 19 18 17 16 15 14 13 12 11 10 Row C 28 27 26 25 24 23 Row C 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 Row C Row D 30 29 28 27 26 25 24 23 Row D 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 Row D Row E 31 30 29 28 27 26 25 24 23 Row E 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 Row E Row F 31 30 29 28 27 26 25 24 23 Row F 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 Row F Row G 31 30 29 28 27 26 25 24 23 Row G 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 Row G Row H 31 30 29 28 27 26 25 24 23 Row H 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 Row H Row I 31 30 29 28 27 26 25 24 23 Row I 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 Row I Row J 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 Row J Row J 31 30 29 28 27 26 25 24 23 23 Row K 22 13 12 11 10 9 8 7 6 5 2 1 Row K 31 30 29 28 27 26 25 24 21 20 19 18 17 16 15 14 4 3 Row K Row L 31 30 29 28 27 26 25 24 23 Row L 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 Row L Row M 31 30 29 28 27 26 25 24 23 Row M 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 Row M Row N 31 30 29 28 27 26 25 24 23 Row N 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 Row N Row O 31 30 29 28 27 26 25 24 23 Row O 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 Row O 23 Row P 9 8 7 6 5 4 3 2 1 Row P 31 30 29 28 27 26 25 24 22 21 20 19 18 17 16 15 14 13 12 11 10 Row P Row Q 31 30 29 28 27 26 25 24 23 Row Q 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 Row Q Row R 31 30 29 28 27 26 25 24 23 Row R 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 Row R table broken desk 9 8 7 6 5 4 3 2 1 Projection Booth
MGMT 276: Statistical Inference in Management Fall, 2014 Welcome Green sheets
Reminder A note on doodling Talking or whispering to your neighbor can be a problem for us – please consider writing short notes.
It went really well! Exam 3 Thanks for your patience and cooperation Grades have been posted
Frequency Score on Exam Remember… In a negatively skewed distribution: mean < median < mode 90 = mode = tallest point 85 = median = middle score 81 = mean = balance point Frequency Note: Always “frequency” Score on Exam Mode Mean Median Note: Label and Numbers
Schedule of readings Before our next exam (December 4th) Lind (10 – 12) Chapter 13: Linear Regression and Correlation Chapter 14: Multiple Regression Chapter 15: Chi-Square Plous (2, 3, & 4) Chapter 17: Social Influences Chapter 18: Group Judgments and Decisions
Exam 4 – Optional Times for Final Two options for completing Exam 4 Thursday (12/4/14) – The regularly scheduled time Tuesday (12/9/14) – The optional later time Must sign up to take Exam 4 on Tuesday (12/2) Only need to take one exam – these are two optional times
Homework due – Tuesday (November 18th) On class website: Please print and complete homework worksheet #18 Hypothesis Testing with ANOVAs
By the end of lecture today 11/13/14 Use this as your study guide By the end of lecture today 11/13/14 Logic of hypothesis testing Steps for hypothesis testing Hypothesis testing with analysis of variance (ANOVA) Interpreting excel output of hypothesis tests Constructing brief, complete summary statements
Study Type 3: One-way ANOVA Single Independent Variable comparing more than two groups Single Dependent Variable (numerical/continuous) Used to test the effect of the IV on the DV Ian was interested in the effect of incentives for girl scouts on the number of cookies sold. He randomly assigned girl scouts into one of three groups. The three groups were given one of three incentives and looked to see who sold more cookies. The 3 incentives were 1) Trip to Hawaii, 2) New Bike or 3) Nothing. This is an example of a true experiment How could we make this a quasi-experiment? Independent Variable: Type of incentive Levels of Independent Variable: None, Bike, Trip to Hawaii Dependent Variable: Number of cookies sold Levels of Dependent Variable: 1, 2, 3 up to max sold Between participant design Causal relationship: Incentive had an effect – it increased sales
Study Type 3: One-way ANOVA Single Independent Variable comparing more than two groups Single Dependent Variable (numerical/continuous) Used to test the effect of the IV on the DV Ian was interested in the effect of incentives for girl scouts on the number of cookies sold. He randomly assigned girl scouts into one of three groups. The three groups were given one of three incentives and looked to see who sold more cookies. The 3 incentives were 1) Trip to Hawaii, 2) New Bike or 3) Nothing. This is an example of a true experiment Dependent variable is always quantitative Sales per Girl scout Sales per Girl scout None New Bike Trip Hawaii None New Bike Trip Hawaii In an ANOVA, independent variable is qualitative (& more than two groups)
One-way ANOVA versus Chi Square Be careful you are not designing a Chi Square If this is just frequency you may have a problem This is an Chi Square Total Number of Boxes Sold Sales per Girl scout This is an ANOVA None New Bike Trip Hawaii None New Bike Trip Hawaii These are just frequencies These are just frequencies These are just frequencies These are means These are means These are means
z scores Review Difference between means z score: A score that indicates how many standard deviations an observation is above or below the mean of the distribution z score = raw score - mean standard deviation Variability of curve(s) Review
“Between Groups” Variability Difference between means . “Between Groups” Variability Difference between means Difference between means “Within Groups” Variability Variability of curve(s) Variability of curve(s) Variability of curve(s) “Between Groups” Variability “Within Groups” Variability
One-way ANOVA One-way ANOVAs test only one independent variable Number of cookies sold One-way ANOVA None Bike 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, 15) = 3.0; p < 0.05 Tells us generally about number of participants / observations
F = MSBetween MSWithin 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 MSWithin MSBetween F = Still, variability of 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
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) 14 9 19 13 15 Troop 2 (bicycle) 12 14 10 11 13 Troop 3 (nada) 10 8 12 7 13 x = 10 x = 14 x = 12 Note: 5 girls in each troop
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 1 (nada) 10 8 12 7 13 Troop 2 (bicycle) 12 14 10 11 13 Troop 3 (Hawaii) 14 9 19 13 15 What is Dependent Variable? How many groups? n = 5 x = 10 n = 5 x = 12 n = 5 x = 14
incentive then the means are significantly different from each other 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) We don’t know which means are different from which …. just that they are not all the same 23
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: Decision rule = .05 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 26
ANOVA table Source df MS F SS Between ? ? ? ? Within ? ? ? Total ? ? “SS” = “Sum of Squares” - will be given for exams - you can think of this as the numerator in a standard deviation formula ANOVA table Source df MS F SS Between ? ? ? ? Within ? ? ? Total ? ?
# scores - number of groups “SS” = “Sum of Squares” - will be given for exams - you can think of this as the numerator in a standard deviation formula ANOVA table Source df MS F SS 3-1=2 Between 40 ? ? 2 # groups - 1 ? ? 15-3=12 Within 88 ? 12 ? # scores - number of groups ? Total 128 ? ? 14 # scores - 1 15- 1=14
ANOVA table SSbetween dfbetween 40 2 40 2 =20 MSbetween MSwithin Source df MS F SS 20 7.33 =2.73 Between 40 2 ? 20 ? 2.73 Within 88 12 7.33 ? Total 128 14 SSwithin dfwithin 88 12 =7.33 88 12
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 problem Using 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 one Replication of study (new data)
Let’s do same problem Using MS Excel
Let’s do same problem Using MS Excel
# scores - number of groups SSbetween dfbetween 40 2 40 2 =20 3-1=2 # groups - 1 MSbetween MSwithin # scores - number of groups 15-3=12 SSwithin dfwithin 88 12 =7.33 20 7.33 =2.73 88 12 # scores - 1 15- 1=14
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.
One way analysis of variance Variance 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 – like within group variability
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 40
F = ANOVA Variability between groups Variability within groups “Between” variability bigger than “within” variability so should get a big (significant) F Variability Within Groups Variability Within Groups Variability Between Groups Variability Within Groups “Between” variability getting smaller “within” variability staying same so, should get a smaller F Variability Between Groups “Between” variability getting very small “within” variability staying same so, should get a very small F Variability Within Groups
ANOVA Variability between groups F = Variability within 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 relative to variability of distributions . Effect size is considered relative to variability of distributions Treatment Effect x Variability between groups Treatment Effect x Variability within groups
Let’s try one In a one-way ANOVA we have three types of variability. Which picture best depicts the random error variability (also known as the within variability)? a. Figure 1 b. Figure 2 c. Figure 3 d. All of the above 1. correct 2. 3.
Which figure would depict the largest F ratio a. Figure 1 b. Figure 2 Variability between groups F = Let’s try one Variability within groups “F ratio” is referring to "observed F” Which figure would depict the largest F ratio a. Figure 1 b. Figure 2 c. Figure 3 d. All of the above correct 1. 2. 3.
Winnie found an observed z of .74, what should she conclude? If your observed z is within one standard deviation of the mean, you will never reject the null Let’s try one Winnie found an observed z of .74, what should she conclude? (Hint: notice that .74 is less than 1) a. Reject the null hypothesis b. Do not reject the null hypothesis c. Not enough info is given correct x x small observed z score small observed z score
Winnie found an observed t of .04, what should she conclude? Let’s try one Winnie found an observed t of .04, what should she conclude? (Hint: notice that .04 is less than 1) a. Reject the null hypothesis b. Do not reject the null hypothesis c. Not enough info is given correct x small observed t score
Let’s try one Winnie found an observed F ratio of .9, what should she conclude? a. Reject the null hypothesis b. Do not reject the null hypothesis c. Not enough info is given correct 1. 2. 3.
F(4, 25) = 3.12; p < 0.05 Let’s try one How many observations within each group? Let’s try one An ANOVA was conducted comparing different types of solar cells and there appears to be a significant difference in output of each (watts) F(4, 25) = 3.12; p < 0.05. In this study there were __ types of solar cells and __ total observations in the whole study? a. 4; 25 b. 5; 30 c. 4; 30 d. 5; 25 F(4, 25) = 3.12; p < 0.05 correct # groups - 1 # scores - # of groups # scores - 1
F(4, 25) = 3.12; p < 0.05 Let’s try one An ANOVA was conducted comparing different types of solar cells and there appears to be significant difference in output of each (watts) F(4, 25) = 3.12; p < 0.05. In this study ___ a. we rejected the null hypothesis b. we did not reject the null hypothesis correct F(4, 25) = 3.12; p < 0.05 Observed F bigger than Critical F p < .05
F(4, 25) = 3.12; p < 0.05 Let’s try one An ANOVA was conducted comparing different types of solar cells. The analysis was completed using an alpha of 0.05. But Julia now wants to know if she can reject the null with an alpha of at 0.01. In this study ___ a. we rejected the null hypothesis b. we did not reject the null hypothesis correct F(4, 25) = 3.12; p < 0.05 Comparison of the Observed F and Critical F Is no longer are helpful because the critical F is no longer correct. We must use the p value p < .05 p > .01
Let’s try one An ANOVA was conducted comparing home prices in four neighborhoods (Southpark, Northpark, Westpark, Eastpark) . For each neighborhood we measured the price of four homes. Please complete this ANOVA table. Degrees of freedom between is _____; degrees of freedom within is ____ a. 16; 4 b. 4; 16 c. 12; 3 d. 3; 12 correct .
Let’s try one An ANOVA was conducted comparing home prices in four neighborhoods (Southpark, Northpark, Westpark, Eastpark) . For each neighborhood we measured the price of four homes. Please complete this ANOVA table. Mean Square between is _____; Mean Square within is ____ a. 300, 300 b. 100, 100 c. 100, 25 d. 25, 100 correct .
Let’s try one An ANOVA was conducted comparing home prices in four neighborhoods (Southpark, Northpark, Westpark, Eastpark) . For each neighborhood we measured the price of four homes. Please complete this ANOVA table. The F ratio is: a. .25 b. 1 c. 4 d. 25 correct .
a. reject the null hypothesis b. not reject the null hypothesis Let’s try one An ANOVA was conducted comparing home prices in four neighborhoods (Southpark, Northpark, Westpark, Eastpark) . For each neighborhood we measured the price of four homes. Please complete this ANOVA table. We should: a. reject the null hypothesis b. not reject the null hypothesis correct Observed F bigger than Critical F p < .05
Let’s try one An ANOVA was conducted comparing home prices in four neighborhoods (Southpark, Northpark, Westpark, Eastpark) . For each neighborhood we measured the price of four homes. The most expensive neighborhood was the ____ neighborhood a. Southpark b. Northpark c. Westpark d. Eastpark correct
An ANOVA was conducted comparing home prices in four neighborhoods (Southpark, Northpark, Westpark, Eastpark) . For each neighborhood we measured the price of four homes. Please complete this ANOVA table. The best summary statement is: a. F(3, 12) = 4.0; n.s. b. F(3, 12) = 4.0; p < 0.05 c. F(3, 12) = 3.49; n.s. d. F(3, 12) = 3.49; p < 0.05 correct
Let’s try one An ANOVA was conducted and there appears to be a significant difference in the number of cookies sold as a result of the different levels of incentive F(2, 27) = ___; p < 0.05. Please fill in the blank a. 3.3541 b. .00635 c. 6.1363 d. 27.00 correct
An ANOVA was conducted and we found the following results: F(3,12) = 3 An ANOVA was conducted and we found the following results: F(3,12) = 3.73 ____. Which is the best summary a. The critical F is 3.89; we should reject the null b. The critical F is 3.89; we should not reject the null c. The critical F is 3.49; we should reject the null d. The critical F is 3.49; we should not reject the null correct Let’s try one
Thank you! See you next time!!