1. URBP 204A QUANTITATIVE METHODS I Statistical 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 between two groups that 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 : H 0 : µ Trips San Jose = µ Trips San Francisco Research : H 1 : Xbar Trips San Jose ≠ Xbar Trips 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 = 5.43Mean = 5.53n = 30 s = 3.42s = 2.06 Trips San JoseTrips San Francisco 7864258556 35103548462 38 5249448 25571258639 35111553727 8419472776

11. t-Test of Independent Samples Steps (Continued) – Computation of obtained test statistic value t obtained = -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= n 1 + n 2 – 2] Consider number of tails (is there directionality?) t critical = 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 t obtained = |-0.14| < t critical = 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 between one group at 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 : H 0 : µ before training = µ after training Research : H 1 : Xbar before training < Xbar after 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 SubjectBeforeAfterDifferenceDifference 2 137416 25839 34624 46711 55839 6594 74624 85611 9374 106824 117811 12871 Sum61872682

18. t-Test of Dependent Samples Steps (Continued) – Computation of obtained test statistic value t obtained = 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?) t critical = 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 t obtained = |4.91| > t critical = 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 between multiple 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 : H 0 : µ group 1 = µ group 2 = µ group 3 Research : H 1 : Xbar group 1 ≠ Xbar group 2 ≠ Xbar group 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 F obtained = 65.31 Degrees of Freedom – Numerator = 2 – Denominator = 27 DataGroup 1 Group 2 Group 3 321 431 521 531 521 411 411 311 411 511 n10 Sum421710 Mean4.21.71.0

27. Simple ANOVA Steps (Continued) – Computation of obtained test statistic value F obtained = 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 F critical = 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 F obtained = 65.31 > F critical = 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.