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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) 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 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 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 t-Tests General Points 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?” 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 : • Research : H0 : µTrips San Jose = µTrips San Francisco H1 : XbarTrips San Jose ≠ XbarTrips San Francisco – Set significance level • Level of risk of Type I Error = 5% • Level of Significance (p) = 0.05 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) 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 t-Test of Independent Samples • Data Trips San Jose Trips San Francisco 7 8 6 4 2 5 8 5 5 6 3 5 10 3 5 4 8 4 6 2 3 8 10 5 2 4 9 4 4 8 2 5 5 7 12 5 8 6 3 9 3 5 1 1 15 5 3 7 2 7 8 4 1 9 4 7 2 7 7 6 San Jose Mean = 5.43 n = 30 s = 3.42 San Francisco Mean = 5.53 n = 30 s = 2.06 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 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 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?” t-Test of Dependent Samples • Example: – “Does sensitivity training make people more sensitive?” • Steps: – State hypotheses • Null : • Research : H0 : µbefore training = µafter training H1 : Xbarbefore training < Xbarafter training – Set significance level • Level of risk of Type I Error = 5% • Level of Significance (p) = 0.05 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) t-Test of Dependent Samples • Formula t-Test of Dependent Samples Subject Before After Difference Difference2 1 3 7 4 16 2 5 8 3 9 3 4 6 2 4 4 6 7 1 1 5 5 8 3 9 6 5 9 4 16 7 4 6 2 4 8 5 6 1 1 9 3 7 4 16 10 6 8 2 4 11 7 8 1 1 12 8 7 -1 1 Sum 61 87 26 82 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 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 Goodbye, t-Tests. Hello, ANOVA. 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?” 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 Simple ANOVA • Example: – “Are there differences in the speech development among three groups of preschoolers?” • Steps: – State hypotheses • Null : • Research : H0 : µgroup 1 = µgroup 2 = µgroup 3 H1 : Xbargroup 1 ≠ Xbargroup 2 ≠ Xbargroup 3 – Set significance level • Level of risk of Type I Error = 5% • Level of Significance (p) = 0.05 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) Simple ANOVA • Formula SumSquaresbetween MeanSumSquaresbetween k 1 F SumSquareswithin MeanSumSquares within N k When : k groups; N cases Simple ANOVA Data Group 1 Group 2 Group 3 3 2 1 4 3 1 5 2 1 5 3 1 5 2 1 4 1 1 4 1 1 3 1 1 4 1 1 5 1 1 n 10 10 10 Sum 42 17 10 Mean 4.2 1.7 1.0 • Fobtained = 65.31 • Degrees of Freedom – Numerator = 2 – Denominator = 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 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.