Question ======== A survey with 51 persons was conducted to analyze the design of an advertising campaign. Each respondent was asked to evaluate the overall impression of the advertisement on an eleven-point scale from 0 (bad) to 10 (good). The evaluations are summarized separately with respect to type of occupation of the respondents in the following figure. ![](boxplots-1.svg) To analyze the influence of occupation on the evaluation of the advertisement an analysis of variance was performed: ``` Res.Df RSS Df Sum of Sq F Pr(>F) 1 50 53.549 2 47 34.018 3 19.531 8.995 8.1265e-05 ``` Which of the following statements are correct? Answerlist ---------- * It can be shown that the evaluation of the respondents depends on their occupation. (Significance level $5$%) * The fraction of explained variance is larger than $60$%. * A one-sided alternative was tested for the mean values. * The fraction of explained variance is smaller than $45$%. * The test statistic is larger than $7.5$. Solution ======== In order to be able to answer the questions the fraction of explained variance has to be determined. The residual sum of squares when using only a single overall mean value ($\mathit{RSS}_0$) as well as the residual sum of squares when allowing different mean values given occupation ($\mathit{RSS}_1$) are required. Both are given in the RSS column of the ANOVA table. The fraction of explained variance is given by $1 - \mathit{RSS}_1/\mathit{RSS}_0 = 1 - 34.018/53.549 = 0.365$. The statements above can now be evaluated as right or wrong. Answerlist ---------- * True. The $p$ value is $8.13e-05$ and hence significant. It can be shown that the evaluations differ with respect to the occupation of the respondents. * False. The fraction of explained variance is $0.365$ and hence _not_ larger than 0.6. * False. An ANOVA always tests the null hypothesis, that all mean values are equal against the alternative hypothesis that they are different. * True. The fraction of explained variance is $0.365$ and hence smaller than 0.45. * True. The test statistic is $F = 8.995$ and hence larger than $7.5$. Meta-information ================ extype: mchoice exsolution: 10011 exname: Analysis of variance