ISYE 6414: Regression Analysis Homework 2: Solutions

I Auto-Gradable: True or False? 1 The assumptions for ANOVA are independence, constant variance, linearity and normality. FALSE 2 The mean sum of squared of errors in ANOVA is the between group variance. FALSE 3 The F ...

I Auto-Gradable: True or False? 1 The assumptions for ANOVA are independence, constant variance, linearity and normality. FALSE 2 The mean sum of squared of errors in ANOVA is the between group variance. FALSE 3 The F-value for testing equality of means using ANOVA is the ratio between mean sum of squared errors and mean sum of squared treatments. FALSE 4 In pairwise comparisons in ANOVA, we need to correct the critical points in the pairwise confidence intervals to correct for multiple inferences. TRUE 5 The sum of squared total is the sum between mean sum of squared errors and mean sum of squared treatments. FALSE 6 The F-value for testing equality of means using ANOVA is the ratio between mean sum of squared errors and mean sum of squared treatments. FALSE 7 We conclude that the means in the ANOVA model are different if the within-group variability is larger than the between-group variability. FALSE 8 The ANOVA is a linear regression model with one qualitative predicting variable. TRUE 9 The sampling distribution for the variance estimator in ANOVA is χ 2 (chi-square) regardless of the assumption of the data. FALSE 10 The number of degrees of freedom of the χ 2 distribution for the variance estimator is N − k where k is the number of samples. TRUE 11 If the F-test for equal means is not rejected we conclude that the means are definitely equal. FALSE 12 If the constant variance assumption in ANOVA does not hold, the inference on the equality of the means will not be reliable. TRUE 13 For assessing the normality assumption of the ANOVA model, we can use the quantilequantile normal plot and the histogram of the residuals. TRUE 14 The linear regression model with a qualitative predicting variable with k levels/classes will have k + 1 parameters to estimate. TRUE 15 If one confidence interval in the pairwise comparison includes only positive values, we conclude that the difference in means is statically positive. TRUE 16 Independence assumption in ANOVA is not essential in inference on equality of the means. FALSE 17 If the difference in the estimated means is very large we definitely conclude that the means are not equal. FLASE 18 We can assess the assumption of constant-variance by plotting the residuals against fitted values. TRUE 1 19 The mean sum of squared errors is the sum of the squared residuals divided by the sample size. FALSE 20 If one confidence interval in the pairwise comparison includes zero, we conclude that the two means are plausibly equal. TRUE II R Example a Consider the following incomplete R output: Source Df Sum of Squares Mean Squares F-statistics p-value Treatments ? ? 116.66 ? 0.0187 Error ? 276.4 ? Total ? ? Find the missing values in the analysis of variance table. N-k = 16-3 = 13 k-1 = 3-1 = 2 N-k + k - 1 = N-1 = 15 MSE*(N-k) = 233.32 SSE + SSTR = 509.72 SSTR / (k-1) = 21.26 SSE/(N−k) SSTR/(k−1) = 5.487 Source Df Sum of Squares Mean Squares F-statistics p-value Treatments 2 233.32 116.66 5.487 0.0187 Error 13 276.4 21.26 Total 15 509.72 b Use µ1, µ2, µ3 as notation for the mean parameters and define these parameters clearly based in the context of the topic above. Find the estimates of these parameters. µ1 : mean difference in IQ for students in group 1 µb1 = 6.2 µ2 : mean difference in IQ for students in group 2 µb2 = 7.83 µ3 : mean difference in IQ for students in group 3 µb3 = 15.2 σ 2 : variance of the differences in IQ σb 2 = MSE = 1516.58 c What are the null and alternative hypotheses of interest in ANOVA? Use the parameters in part (b) to write the two hypothesis. Based on the ANOVA table in part (a), answer the following questions: H0 : µ1 = µ2 = µ3 HA : at least 2 of the means are not equal (µ1 6= µ2 and/or µ1 6= µ3 and/or µ2 6= µ3)