URBP 204A QUANTITATIVE METHODS I Statistical Analysis Lecture III

1. URBP 204A QUANTITATIVE METHODS IStatistical Analysis Lecture III Gregory Newmark San Jose State University (This lecture accords with Chapters 9,10, & 11 of Neil Salkind’s Statistics for People who (Think They) Hate Statistics)

2. Statistical Significance Revisited Steps: • State hypothesis • Set significance level associated with null hypothesis • Select statistical test (we will learn these soon) • Computation of obtained test statistic value • Computation of critical test statistic value • Comparison of obtained and critical values • If obtained > critical reject the null hypothesis • If obtained < critical stick with the null hypothesis

3. Three Statistical Tests t-Test for Independent Samples • Tests between the means of two different groups t-Test for Dependent Samples • Tests between the means of two related groups Analysis of Variance (ANOVA) • Tests between means of more than two groups

4. t-Tests General Points Used for comparing sample means when population’s standard deviation is unknown (which is almost always) Accounts for the number of observations Distribution of t-statistic is identical to normal distribution when sample sizes exceed 120

5. t-Tests General Points

6. t-Test of Independent Samples Compares observations of a single variable betweentwo groupsthat are independent Examples: • “Are there differences in TV exposure between teens in Oakland and San Francisco?” • “We are going to take 100 people and give 50 of them $2 and see which group is happier.” • “In 2008, did the average visitor spend less time at the art museum than at the planetarium?” • “Do people in San Jose make different amounts of monthly transit trips than folks in San Francisco?”

7. t-Test of Independent Samples Example: • “Do people in San Jose make different amounts of monthly transit trips than folks in San Francisco?” Steps: • State hypotheses • Null : H0 : µTrips San Jose = µTripsSan Francisco • Research : H1 : XbarTrips San Jose ≠ XbarTrips San Francisco • Set significance level • Level of risk of Type I Error = 5% • Level of Significance (p) = 0.05

8. t-Test of Independent Samples Steps (Continued) • Select statistical test • t-Test of Independent Samples • Computation of obtained test statistic value • Insert obtained data into appropriate formula • (SPSS can expedite this step for us)

9. t-Test of Independent Samples Formula Where • Xbar is the mean • n is the number of participants • s is the standard deviation • Subscripts distinguish between Groups 1 and 2

10. t-Test of Independent Samples Data San JoseSan Francisco Mean = 5.43 Mean = 5.53 n = 30 n = 30 s = 3.42 s = 2.06

11. t-Test of Independent Samples Steps (Continued) • Computation of obtained test statistic value • tobtained = -0.14 • (don’t worry about the sign here) • Computation of critical test statistic value • Value needed to reject null hypothesis • Look up p = 0.05 in t table • Consider degrees of freedom [df= n1 + n2 – 2] • Consider number of tails (is there directionality?) • tcritical = 2.001

12. t-Test of Independent Samples Steps (Continued) • Comparison of obtained and critical values • If obtained > critical reject the null hypothesis • If obtained < critical stick with the null hypothesis • tobtained = |-0.14| < tcritical = 2.001 • Therefore, we cannot reject the null hypothesis and we thus conclude that there are no differences in the mean transit trips per month between people in San Jose and San Francisco

13. t-Test of Dependent Samples Compares observations of a single variable betweenone groupat two time periods Examples: • “Does watching this movie make audiences feel happier?” • “Does a certain curriculum initiative improve student test results?” • “Do people make more transit trips with the extension of a BART line to their neighborhood?” • “Does sensitivity training make people more sensitive?”

14. t-Test of Dependent Samples Example: • “Does sensitivity training make people more sensitive?” Steps: • State hypotheses • Null : H0 : µbefore training = µafter training • Research : H1 : Xbarbefore training < Xbarafter training • Set significance level • Level of risk of Type I Error = 5% • Level of Significance (p) = 0.05

15. t-Test of Dependent Samples Steps: • Select statistical test • t-Test of Dependent Samples • Computation of obtained test statistic value • Insert obtained data into appropriate formula • (SPSS can expedite this step for us)

16. t-Test of Dependent Samples Formula

17. t-Test of Dependent Samples

18. t-Test of Dependent Samples Steps (Continued) • Computation of obtained test statistic value • tobtained = 4.91 • (don’t worry about the sign here) • Computation of critical test statistic value • Value needed to reject null hypothesis • Look up p = 0.05 in t table • Consider degrees of freedom [df = n -1 ] • Consider number of tails (is there directionality?) • tcritical = 1.80

19. t-Test of Dependent Samples Steps (Continued) • Comparison of obtained and critical values • If obtained > critical reject the null hypothesis • If obtained < critical stick with the null hypothesis • tobtained = |4.91| > tcritical = 1.80 • Therefore, we reject the null hypothesis and we thus conclude that the sensitivity training works

20. Goodbye, t-Tests. Hello, ANOVA.

21. Simple ANOVA Compares observations of a single variable betweenmultiple groups Examples: • “Are there differences between the reading skills of high school, college, and graduate students?” • “Does environmental knowledge vary between people who commute by car, bus, and walking?” • “Are there wealth differences between A’s, Giants, Dodger, and Angels fans?” • “Are there differences in the speech development among three groups of preschoolers?”

22. Simple ANOVA Also called One-way ANOVA Compares means of more than two groups on one factor or dimension with F statistic Calculated as a ratio of the amount of variability between groups (due to the grouping factor) to the amount of variability within groups (due to chance) • F = Variability between different Groups Variability within each Group • As this ratio exceeds one it is more likely to be due to something other than chance No directionality, therefore no issue of tails

23. Simple ANOVA Example: • “Are there differences in the speech development among three groups of preschoolers?” Steps: • State hypotheses • Null : H0 : µgroup 1 = µgroup 2 = µgroup 3 • Research : H1 : Xbargroup 1 ≠ Xbargroup 2 ≠ Xbargroup 3 • Set significance level • Level of risk of Type I Error = 5% • Level of Significance (p) = 0.05

24. Simple ANOVA Steps: • Select statistical test • Simple ANOVA • Computation of obtained test statistic value • Insert obtained data into appropriate formula • (SPSS can expedite this step for us)

25. Simple ANOVA Formula

26. Simple ANOVA Fobtained = 65.31 Degrees of Freedom • Numerator = 2 • Denominator = 27

27. Simple ANOVA Steps (Continued) • Computation of obtained test statistic value • Fobtained = 65.31 • Computation of critical test statistic value • Value needed to reject null hypothesis • Look up p = 0.05 in F table • Consider degrees of freedom for numerator and denominator • No need to worry about number of tails • Fcritical = 3.36

28. Simple ANOVA Steps (Continued) • Comparison of obtained and critical values • If obtained > critical reject the null hypothesis • If obtained < critical stick with the null hypothesis • Fobtained = 65.31 > Fcritical = 3.36 • Therefore, we reject the null hypothesis and we thus conclude that there are differences in the speech abilities of the students in the preschools.