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F-Test) CONTINUOUS T-Test ANOVA -Correlation (F-Test) -Simple Linear Regression

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F-Test) CONTINUOUS T-Test ANOVA -Correlation (F-Test) -Simple Linear Regression Bivariate Analysis Variable 1 2 LEVELS >2 LEVELS CONTINUOUS Variable 2 Variable 2 LEVELS X2 X2 t-test chi square test chi square test >2 LEVELS X2 X2 ANOVA chi square test chi square test (F-test) CONTINUOUS t-test ANOVA -Correlation (F-test) -Simple linear Regression Comparison of means: F-test Example 1: Research question: Among university students, is the average weight of students in university “A” different than that in university “B” and that in university “C”? Is there an association between weight and type of university? Ho : Average weight A = Average weight B = Average weight C Ha : At least two averages are different Statistical test: F-test = (Analysis of Variance)= ANOVA Comparison of means: F-test One way F-Test (SPSS output): Example 1 Descriptives weight 95% Confidence Interval for Mean N Mean Std. Deviation Std. Error Lower Bound Upper Bound Minimum Maximum A 290 65.59 13.297 .781 64.06 67.13 41 125 B 1340 63.46 14.201 .388 62.70 64.22 39 135 C 345 67.74 15.299 .824 66.12 69.36 42 115 Total 1975 64.52 14.360 .323 63.89 65.15 39 135 ANOVA weight Sum of Squares df Mean Square F Sig. Between Groups 5414.963 2 2707.482 13.293 .000 Within Groups 401651.5 1972 203.677 Total 407066.5 1974 Comparison of means: F-test This is the p-value for the F-test (testing of the null hypothesis of whether the mean of weight for A = mean of weight for B = mean of weight for C). If this p-value is > 0.05 then accept null hypothesis and conclude that the means of the 3 groups are equal. If the p-value is < 0.05 then reject null hypothesis (accept the alternative) and conclude that at least two means are different. ANOVA weight Sum of Squares df Mean Square F Sig. Between Groups 5414.963 2 2707.482 13.293 .000 Within Groups 401651.5 1972 203.677 Total 407066.5 1974 Comparison of means: F-test ANOVA weight Sum of Squares df Mean Square F Sig. Between Groups 5414.963 2 2707.482 13.293 .000 Within Groups 401651.5 1972 203.677 Total 407066.5 1974 Since p-value is < 0.05 then reject null hypothesis (accept the alternative) and conclude that at least two means are different. BUT which of the means are different??? Comparison of means: F-test If we want to know exactly what 2 means are different: need to ask for Post Hoc Test Post Hoc Tests Multiple Comparisons Dependent Variable: weight Bonferroni Mean Difference 95% Confidence Interval (I) university (J) university (I-J) Std. Error Sig. Lower Bound Upper Bound A B 2.135 .924 .063 -.08 4.35 C -2.144 1.137 .178 -4.87 .58 B A -2.135 .924 .063 -4.35 .08 C -4.279* .862 .000 -6.34 -2.21 C A 2.144 1.137 .178 -.58 4.87 B 4.279* .862 .000 2.21 6.34 *. The mean difference is significant at the .05 level. Comparison of means: F-test A p-value < 0.05 (*) identifies significance between 2 groups: In this example differences in average of weight are between B and C. Post Hoc Tests Multiple Comparisons Dependent Variable: weight Bonferroni Mean Difference 95% Confidence Interval (I) university (J) university (I-J) Std. Error Sig. Lower Bound Upper Bound A B 2.135 .924 .063 -.08 4.35 C -2.144 1.137 .178 -4.87 .58 B A -2.135 .924 .063 -4.35 .08 C -4.279* .862 .000 -6.34 -2.21 C A 2.144 1.137 .178 -.58 4.87 B 4.279* .862 .000 2.21 6.34 *. The mean difference is significant at the .05 level. Comparison of means: F-test Example 1: Research question: Is there an association between weight and type of university? Ho : Average weight A = Average weight B = Average weight C Ha : At least two averages are different Statistical test: F-test = 13.293; p<0.05 Conclusion: There is a significant relationship between weight and type of university. Based on the post Hoc test, differences in average of weight are between B and C. Comparison of means: F-test One way F-Test (SPSS output): Example 2 Descriptives height 95% Confidence Interval for Mean N Mean Std. Deviation Std. Error Lower Bound Upper Bound Minimum Maximum A 291 170.63 12.197 .715 169.22 172.03 72 194 B 1339 169.47 9.446 .258 168.96 169.97 58 202 C 341 171.80 9.336 .506 170.80 172.79 150 201 Total 1971 170.04 9.917 .223 169.60 170.48 58 202 ANOVA height Sum of Squares df Mean Square F Sig. Between Groups 1589.713 2 794.856 8.140 .000 Within Groups 192172.9 1968 97.649 Total 193762.6 1970 Comparison of means: F-test Post Hoc Tests (SPSS output): Example 2 Multiple Comparisons Dependent Variable: height Bonferroni Mean Difference 95% Confidence Interval (I) university (J) university (I-J) Std. Error Sig. Lower Bound Upper Bound A B 1.158 .639 .211 -.37 2.69 C -1.171 .789 .413 -3.06 .72 B A -1.158 .639 .211 -2.69 .37 C -2.328* .599 .000 -3.76 -.89 C A 1.171 .789 .413 -.72 3.06 B 2.328* .599 .000 .89 3.76 *. The mean difference is significant at the .05 level. Comparison of means: F-test One way F-Test (SPSS output): Example 2 Descriptives height 95% Confidence Interval for Mean N Mean Std. Deviation Std. Error Lower Bound Upper Bound Minimum Maximum A 291 170.63 12.197 .715 169.22 172.03 72 194 B 1339 169.47 9.446 .258 168.96 169.97 58 202 C 341 171.80 9.336 .506 170.80 172.79 150 201 Total 1971 170.04 9.917 .223 169.60 170.48 58 202 Research question: Is there an association between height and type of university? Comparison of means: F-test One way F-Test (SPSS output): Example 2 Descriptives height 95% Confidence Interval for Mean N Mean Std. Deviation Std. Error Lower Bound Upper Bound Minimum Maximum A 291 170.63 12.197 .715 169.22 172.03 72 194 B 1339 169.47 9.446 .258 168.96 169.97 58 202 C 341 171.80 9.336 .506 170.80 172.79 150 201 Total 1971 170.04 9.917 .223 169.60 170.48 58 202 Ho: Average height A = Average height B = Average height C Ha: At least two averages are different Comparison of means: F-test One way F-Test (SPSS output): Example 2 ANOVA height Sum of Squares df Mean Square F Sig. Between Groups 1589.713 2 794.856 8.140 .000 Within Groups 192172.9 1968 97.649 Total 193762.6 1970 Value of statistical test: 8.140 P-value: 0.000 Comparison of means: F-test One way F-Test (SPSS output): Example 2 Multiple Comparisons Dependent Variable: height Bonferroni Mean Difference 95% Confidence Interval (I) university (J) university (I-J) Std. Error Sig. Lower Bound Upper Bound A B 1.158 .639 .211 -.37 2.69 C -1.171 .789 .413 -3.06 .72 B A -1.158 .639 .211 -2.69 .37 C -2.328* .599 .000 -3.76 -.89 C A 1.171 .789 .413 -.72 3.06 B 2.328* .599 .000 .89 3.76 *. The mean difference is significant at the .05 level. There is a difference in average of height between B and C. Comparison of means: F-test One way F-Test (SPSS output): Example 2 ANOVA height Sum of Squares df Mean Square F Sig. Between Groups 1589.713 2 794.856 8.140 .000 Within Groups 192172.9 1968 97.649 Total 193762.6 1970 Conclusion: There is a significant relationship between height and type of university. Based on the post Hoc test, differences in average of height are between B and C. SPSS commands for F-test Example 1 Analyze Compare Means One way ANOVA select weight as the dependent variable select university as the factor Go to options- chose descriptive statistics Go to Post Hoc- Select Bonferroni for equal variance assumed SPSS commands for F-test Example 2 Analyze Compare Means One way ANOVA select height as the dependent variable select university as the factor Go to options- chose descriptive statistics Go to Post Hoc- Select Bonferroni for equal variance assumed T-TEST: CI VS. P-VALUE Hypothesis Testing (P-Value) & Confidence Interval (CI) In hypothesis testing (p-value): Decision of whether to accept or not the null hypothesis In confidence interval: Estimation of the parameter Decision of whether to accept or not the null hypothesis P-value & Confidence interval Example 1: T-test Group Statistics Std. Error gender N Mean Std. Deviation Mean weight male 804 75.92 12.843 .453 female 1135 56.47 8.923 .265 Independent Samples Test Levene's Test for Equality of Variances t-test for Equality of Means 95% Confidence Interval of the Mean Std. Error Difference F Sig. t df Sig. (2-tailed) Difference Difference Lower Upper weight Equal variances 132.258 .000 39.337 1937 .000 19.444 .494 18.475 20.414 assumed Equal variances 37.059 1335.508 .000 19.444 .525 18.415 20.473 not assumed CI: Estimation of the parameter Example 1: T-test Independent Samples Test Levene's Test for Equality of Variances t-test for Equality of Means 95% Confidence Interval of the Mean Std.
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