Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Difference Between Means Test (“t” statistic) Analysis of Variance (“F” statistic) Review of test statistics Procedure Level of Measurement Statistic Interpretation Correlation All variables continuous r Range -1 to +1, with 0 meaning no relationship. For example, .35 denotes a moderately strong positive relationship Regression All variables continuous r2, R2 Proportion of change in the dependent variable accounted for by change in the independent variable. R2 denotes cumulative effect of multiple independent variables. Unit change in the dependent variable caused by a one-unit change in the independent variable b Logistic regression DV nominal & dichotomous, IV’s nominal or continuous b exp(B) Don’t try Odds that DV will change if IV changes one unit, or, if IV is dichotomous, if it changes its state. Range 0 to infinity; 1 denotes even odds, or no relationship. Higher than 1 means positive relationship, lower negative relationship. Use percentage to describe likelihood of effect. Chi-Square All variables categorical, not ordinal X2 Reflects difference between Observed and Expected frequencies. Use table to determine if coefficient is sufficiently large to reject null hypothesis Difference between means IV dichotomous, DV continuous t Reflects magnitude of difference. Use table to determine if coefficient is sufficiently large to reject null hypothesis. Difference Between Means Test • Used to test hypotheses with categorical independent variable and continuous dependent variable – Males are taller than females (height in inches) – Female officers are less cynical than male officers (cynicism on a 1-5 scale) • To determine if variables are associated compare the means of two randomly drawn samples • Null hypothesis: More than 5 chances in 100 (p> .05) that the difference between means was produced by chance – To overcome the null we need a difference sufficiently large so that the probability it was produced by chance is less than five in one-hundred (p< .05) – In other words, the t coefficient exceeds sampling error – In the t this sampling error is the “standard error of the difference between means” – the difference between all possible pairs of means, due to chance alone Major advantage: Remember our concern that in large datasets, weak real-life effects can produce statistically significant results? – When comparing means, we know their actual values. This lets us recognize situations when, statistical significance or not, the actual differences are trivial. • Class exercise H1: Male officers more cynical than females (1 - tailed) H2: Officer gender determines cynicism (2 - tailed) 1. 2. 3. 4. 5. Review: What is the null hypothesis? Draw one sample of male officers, one of females Compute the Standard Error of the Difference Between Means Calculate the t coefficient Use the t table to check for significance (no more than 5 chances in 100 that the null hypo. is correct) Note One-tailed hypotheses (direction of the effect on the dependent variable is predicted) require a smaller t to reach statistical significance than twotailed hypotheses, where only an effect is predicted, not its direction Why? Because we’re using only one side of the probability distribution Calculating t 1. Obtain the “pooled sample variance” Sp2 (Simplified method – midpoint between the two sample variances) 2. Compute the S.E. of the Diff. Between Means x1 -x2 3. Compute the statistic t= x 1 -x2 x 1 Sp 2 = s2 1 + s2 2 2 - x2 = Sp2 ( 1 n1 + 1 n2 ) Actual (“obtained”) difference between means Predicted difference due to sampling error • The result is a ratio: the smaller the predicted error, and the greater the obtained difference between means, the larger the t coefficient • The larger the t, the more likely we are to reject the null hypothesis, that any difference between means is due to chance alone • We use a table to determine whether the t is large enough to reject the null hypothesis (see next slide). We can reject the null if the probability that the difference between means is due to chance is less than five in one-hundred (p< .05). • If the probability that the difference between the means is due to chance is greater than five in one-hundred (p> .05), the null hypothesis is true. 1. Is hypothesis one-tailed (direction of change in the DV predicted) or two-tailed (direction not predicted)? H1: Males more cynical than females. This is one-tailed, so use the top row. H2: Males and females differ in cynicism. This is two-tailed, so use the second row. 2. df, “Degrees of Freedom” represents sample size – add the numbers of cases in both samples, then subtract two: df = (n1 + n2) – 2 3. To call a t “significant” (thus reject the null hypothesis) the coefficient must be as large or larger than what is required at the .05 level; that is, we cannot take more than 5 chances in 100 that the difference between means is due to chance. • For a one-tailed test, use the top row, then slide over to the .05 column. For a two-tailed test, use the second row, then slide to .05 column. If the t is smaller than the number at the intersection of the .05 column and the appropriate df row, it is non-significant. • If the t is that size or larger, it is significant. Slide to the right to see if it is large enough to be significant at a more stringent level. More complex mean comparisons: Analysis of Variance When there are more than two groups: Analysis of Variance Independent variables: categorical Dependent variable: continuous Example: does officer professionalism vary between cities? (scale 1-10) City L.A. S.F. S.D. Mean 8 5 3 Calculate the “F” statistic, look up the table. An “F” statistic that is sufficiently large can overcome the null hypothesis that the differences between the means are due to chance. “Two-way” Analysis of Variance • Stratified independent variable(s) • City L.A. S.F. S.D. Mean – M 10 7 5 Mean - F 6 3 2 Within Between F statistic is a ratio of “between-group” to “within” group differences. To overcome the null hypothesis, the differences in scores between groups (between cities and, overall, between genders) should be much greater than the differences in scores within cities Between group variance (error + systematic effects of ind. variable) Within group variance (how scores disperse within each city) Parking lot exercise Homework Homework assignment Two random samples of 10 patrol officers from the XYZ Police Department, each officer tested for cynicism (continuous variable, scale 1-5) Sample 1 scores: 3 3 3 3 3 3 3 1 2 5 -- Variance = .99 Sample 2 scores: 2 1 1 2 3 3 3 3 4 2 -- Variance = .93 Pooled sample variance Sp2 Simplified method: midpoint between the two sample variances 2 Sp = s2 1 + s2 2 2 Standard error of the difference between means x 1 -x2 = Sp2 ( 1 n1 x1 -x2 1 +n ) 2 T-Test for significance of the difference between means x1 -x2 t = -------------x -x 1 2 CALCULATIONS Pooled sample variance: .96 Standard error of the difference between means: .44 t statistic: 1.14 df – degrees of freedom: (n1 + n2) – 2 = 18 Would you use a ONE-tailed t-test OR a TWO-tailed ttest? Depends on the hypothesis Two-tailed (does not predict direction of the change): Gender cynicism One-tailed (predicts direction of the change): Males more cynical than females Can you reject the NULL hypothesis? (probability that the t coefficient could have been produced by chance must be less than five in a hundred) NO – For a ONE-tailed test need a t of 1.734 or higher NO – For a TWO-tailed test need a t of 2.101 or higher Final exam practice • You will be given scores and variances for two samples and asked to decide whether their means are significantly different. • You will be asked to state the null hypothesis. You will then compute the t statistic. You be given formulas, but should know the methods by heart. Please refer to week 15 slide show. • To compute the t you will compute the pooled sample variance and the standard error of the difference between means. • You will then compute the degrees of freedom (adjusted sample size) and use the t table to determine whether the coefficient is sufficiently large to reject the null hypothesis. – Print and bring to class: http://www.sagepub.com/fitzgerald/study/materials/appendices/app_f.pdf – Use the one-tailed test if the direction of the effect is specified, or two-tailed if not • You will be asked to express using words what the t-table conveys about the significance (or non-significance) of the t coefficient • Sample question: Are male CJ majors significantly more cynical than female CJ majors? We randomly sampled five males and five females. Males: 4, 5, 5, 3, 4 Females: 4, 3, 4, 4, 5 – Null hypothesis: No significant difference between cynicism of males and females – Variance for males (provided): 0.7 Variance for females (provided): 0.5 – Pooled sample variance = .6 SE of the difference between means = .49 t = .41 df = 8 – Check the “t” table. Can you reject the null hypothesis? NO – Describe conclusion using words: The t must be at least 1.86 (one-tailed test) to reject the null hypothesis of no significant difference in cynicism, with only five chances in 100 that it is true.