Introduction to Statistics for the Social Sciences SBS200 - Lecture Section 001, Fall 2017 Room 150 Harvill Building 10:00 - 10:50 Mondays, Wednesdays & Fridays. Welcome
Lecturer’s desk Projection Booth Screen Screen Harvill 150 renumbered Row A 15 14 Row A 13 12 11 10 9 8 7 6 5 4 3 2 1 Row A Row B 23 22 21 20 Row B 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 Row B Row C 25 24 23 22 21 Row C 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 Row C Row D 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 1 Row D Row E 31 30 29 28 27 26 25 24 23 Row E 23 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 35 34 33 32 31 30 29 28 27 26 Row F 25 24 23 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 35 34 33 32 31 30 29 28 27 26 Row G 25 24 23 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 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 Row H 12 11 10 9 8 7 6 5 4 3 2 1 Row H 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 Row J 13 12 11 10 9 8 7 6 5 4 3 2 1 Row J 41 40 39 38 37 36 35 34 33 32 31 30 29 Row K 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 Row K Row L 33 32 31 30 29 28 27 26 25 Row L 24 23 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 21 20 19 Row M 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 Row M Row N 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 Row P 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 Harvill 150 renumbered table 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 Projection Booth Left handed desk
A note on doodling
Lab sessions Continue Project 3 This Week Everyone will want to be enrolled in one of the lab sessions Continue Project 3 This Week
Schedule of readings Before next exam (November 17th) Please read chapters 1 - 11 in OpenStax textbook Please read Chapters 2, 3, and 4 in Plous Chapter 2: Cognitive Dissonance Chapter 3: Memory and Hindsight Bias Chapter 4: Context Dependence
Question: Using nicknames…. what is formula for SAMPLE variance? Sum of Squares Degrees of freedom SS df “SS” = “Sum of Squares” “SS” = “Sum of Squares” “df” = degrees of freedom “SS” = “Sum of Squares” We lose one degree of freedom for every parameter we estimate Remember, you should know these four formulas by heart
This is our sample variance… what is formula for SAMPLE variance? SS df ANOVA table Source df MS F SS Between 40 ? ? ? Within 88 ? ? Total 128 ?
Re-dux 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 SS df = MS 4. Complete this ANOVA table ANOVA table 5. Given a critical F(2,12) = 3.89 Write a summary statement of findings (all three parts) Source SS df MS F Between 40 ? ? ? Within 88 ? ? Total 128 ? Re-dux
Re-dux ANOVA table Source df MS F SS Between 40 ? ? 2 ? ? Within 88 ? “SS” = “Sum of Squares” - will be given for exams ANOVA table Source df MS F SS Between 40 ? ? 2 # groups - 1 ? ? 3-1=2 15-3=12 Within 88 ? 12 ? ? # scores - number of groups Total 128 ? ? 14 # scores - 1 15- 1=14 Re-dux
“SS” = “Sum of Squares” - will be given for exams ANOVA table SSbetween dfbetween “SS” = “Sum of Squares” - will be given for exams 40 2 40 2 =20 MSbetween MSwithin ANOVA table 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 Re-dux
Re-dux 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 main 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 main effect of the number of cookies sold as a result of the different levels of incentive F(2, 12) = 2.73; n.s. Re-dux
Hand in your writing assignment
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 15
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 SSbetween “Sum of Squares” 40 =20 dfbetween 2 MSbetween MSwithin # groups - 1 20 7.33 =2.73 3-1=2 88 12 =7.33 # scores - # of groups SSwithin dfwithin 15-3=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?) “Sum of Squares”
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.
Homework
Homework
Homework
Homework Type of major in school 4 (accounting, finance, hr, marketing) Grade Point Average Homework 0.05 2.83 3.02 3.24 3.37
# scores - number of groups 0.3937 0.1119 If observed F is bigger than critical F: Reject null & Significant! If observed F is bigger than critical F: Reject null & Significant! 0.3937 / 0.1119 = 3.517 Homework 3.517 3.009 If p value is less than 0.05: Reject null & Significant! 3 24 0.03 4-1=3 # groups - 1 # scores - number of groups 28 - 4=24 # scores - 1 28 - 1=27
Yes Homework F (3, 24) = 3.517; p < 0.05 The GPA for four majors was compared. The average GPA was 2.83 for accounting, 3.02 for finance, 3.24 for HR, and 3.37 for marketing. An ANOVA was conducted and there is a significant difference in GPA for these four groups (F(3,24) = 3.52; p < 0.05).
Average for each group (We REALLY care about this one) Number of observations in each group
Number of groups minus one (k – 1) 4-1=3 “SS” = “Sum of Squares” - will be given for exams Number of people minus number of groups (n – k) 28-4=24
SS between df between MS between MS within SS within df within
Type of executive 3 (banking, retail, insurance) Hours spent at computer 0.05 10.8 8 8.4
11.46 2 If observed F is bigger than critical F: Reject null & Significant! If observed F is bigger than critical F: Reject null & Significant! 11.46 / 2 = 5.733 5.733 3.88 If p value is less than 0.05: Reject null & Significant! 2 12 0.0179
Yes F (2, 12) = 5.73; p < 0.05 The number of hours spent at the computer was compared for three types of executives. The average hours spent was 10.8 for banking executives, 8 for retail executives, and 8.4 for insurance executives. An ANOVA was conducted and we found a significant difference in the average number of hours spent at the computer for these three groups , (F(2,12) = 5.73; p < 0.05).
Number of observations in each group Average for each group Number of observations in each group Just add up all scores
Number of groups minus one (k – 1) 3-1=2 “SS” = “Sum of Squares” - will be given for exams Number of people minus number of groups (n – k) 15-3=12
MS between MS within SS between df between SS within df within
“Between Groups” Variability Difference between means . “Between Groups” Variability Difference between means Difference between means Difference between means “Within Groups” Variability Variability of curve(s) Variability of curve(s) Variability of curve(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
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
The sum of squared deviations of some set of scores about their mean Sum of squares (SS): The sum of squared deviations of some set of scores about their mean Mean squares (MS): The sum of squares divided by its degrees of freedom Mean square between groups: sum of squares between groups divided by its degrees of freedom Mean square total: sum of squares total divided by its degrees of freedom MSWithin MSBetween F = Mean square within groups: sum of squares within groups divided by its degrees of freedom 45
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)
In a one-way ANOVA we have three types of variability. 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.
In a one-way ANOVA we have three types of variability. Let’s try one In a one-way ANOVA we have three types of variability. Which picture best depicts the between group variability? a. Figure 1 b. Figure 2 c. Figure 3 d. All of the above correct 1. 2. 3.
Questions?
Thank you! See you next time!!