Introduction to Statistics for the Social Sciences SBS
Introduction to Statistics for the Social Sciences SBS 200 - Lecture Section 001, Spring 2017 Room 150 Harvill Building 9: 00 - 9: 50 Mondays, Wednesdays & Fridays. http: //www. youtube. com/watch? v=o. SQJP 40 Pc. GI
A note on doodling
By the end of lecture today 3/31/17 Hypothesis testing with ANOVA
Before next exam (April 7 th) Please read chapters 1 - 11 in Open. Stax textbook Please read Chapters 2, 3, and 4 in Plous Chapter 2: Cognitive Dissonance Chapter 3: Memory and Hindsight Bias Chapter 4: Context Dependence
Homework Assignment 21 Please complete this homeworksheet ANOVA Due: Friday, March 31 st
Homework Assignment 22 Go to D 2 L - Click on “Content” Click on “Interactive Online Homework Assignments” Complete Assignment 22: Ø HW 22 -Part 1 -Completing t-tests with Excel Ø HW 22 -Part 2 -Completing ANOVAs with Excel Due: Monday, April 3 rd Please note: Due date for Assignment 21 (worksheet for ANOVA) has been extended to April 3 rd
Everyone will want to be enrolled in one of the lab sessions Project 3 this week
“SS” = “Sum of Squares” “SS” “Sum ofof. Squares” “n” ==number scores “SS” = “Sum of Squares” “SS”= =degrees “Sum ofof. Squares” “df” freedom Remember, you should know these four formulas by heart We lose one degree of freedom for every parameter we estimate
Standard deviation (definitional formula) - Let’s do one This numerator is called “sum of squares” Step 1: Find the mean ΣX = 45 ΣX / N = 45/9 = 5 Step 2: Subtract the mean from each score Step 3: Square the deviations Step 4: Find standard deviation This is the Variance! a) 60 / 9 = 6. 6667 Each of these are deviation scores 2 _ X_ _ X - µ _ 1 2 3 4 5 6 7 8 9 1 -5=-4 2 -5=-3 3 -5=-2 4 -5=-1 5 -5= 0 6 -5= 1 7 -5= 2 8 -5= 3 9 -5= 4 45 0 b) square root of 6. 6667 = 2. 5820 This is the standard deviation! (X - µ) 16 9 4 1 0 1 4 9 16 60 Σ(x - µ) = 0 Re w e i v
Remember, you should know these four formulas by heart
“SS” = “Sum of Squares” “SS” “Sum ofof. Squares” “n” ==number scores “SS” = “Sum of Squares” “SS”= =degrees “Sum ofof. Squares” “df” freedom Remember, you should know these four formulas by heart We lose one degree of freedom for every parameter we estimate
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 Re w e i v
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 The three primary differences Tells us between generally t-tests and ANOVAS Tells us generally about number of 1. ANOVAs can test more than two means about number of participants / groups / levels of IV by participants / observations 2. We are comparing sample means indirectly observations 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 are: Re w e i v
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 Dependent 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 n=5 x = 10 n=5 x = 12 n=5 x = 14 What is Independent Variable? How many groups? Re w e i v
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) = 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 SS df MS F Between 40 ? Within ? 88 Total ? 128 ? ? ? ? Source
Writing Assignment - ANOVA 1. Write formula for standard deviation of sample 2. Write formula for variance of sample 3. Re-write formula for variance of sample using the nicknames for the numerator and denominator 4. Complete this ANOVA table SS = MS df ANOVA table Source Between Within Total SS df MS F 40 88 128 ? ? ? ?
“SS” = “Sum of Squares” - will be given for exams Source Between Within Total ANOVA table SS df 40 ? 88 ? ? 128 ? 2 ? 14 F MS # groups - 1 ? ? 3 -1=2 ? 15 -3=12 # scores - number of groups # scores - 1 15 - 1=14
ANOVA table 40 SSbetween 2 ANOVA table dfbetween “SS” = “Sum of Squares” - will be given for exams Source Between Within Total SS df 40 88 128 2 12 14 88 12 MS 20 ? ? 7. 33 SSwithin dfwithin 40 =20 2 F ? 2. 73 88 =7. 33 12 MSbetween MSwithin 20 =2. 73 7. 33
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.
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 We don’t know which means are different from which …. just that they are not all the same To get an effect we want: • Large “F” - big effect and small variability • Small “p” - less than 0. 05 (whatever our alpha is)
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
ANOVA table “Sum of Squares” SSbetween 40 =20 2 dfbetween # groups - 1 20 =2. 73 7. 33 3 -1=2 # scores - # of groups 15 -3=12 # scores - 1 15 - 1=14 MSbetween MSwithin SSwithin dfwithin 88 =7. 33 12
No, so it is not significant Do not reject null “Sum of Squares” F critical P-value (is it less than. 05? ) (is observed F greater than critical F? )
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. 7272 Critical F(2, 12) = 3. 88529 F(2, 12) = 2. 73; n. s. 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.
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