Exam 1

  1. Question

    A survey with 49 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.

    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     48 24.789                           
    2     45 24.642  3     0.147 0.089 0.96565
    

    Which of the following statements are correct?


    1. A one-sided alternative was tested for the mean values.
    2. It can be shown that the evaluation of the respondents depends on their occupation. (Significance level 55%)
    3. The fraction of explained variance is larger than 5959%.
    4. The fraction of explained variance is smaller than 5656%.
    5. The test statistic is smaller than 1.21.2.

    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 (𝑅𝑆𝑆0\mathit{RSS}_0) as well as the residual sum of squares when allowing different mean values given occupation (𝑅𝑆𝑆1\mathit{RSS}_1) are required. Both are given in the RSS column of the ANOVA table. The fraction of explained variance is given by 1𝑅𝑆𝑆1/𝑅𝑆𝑆0=124.642/24.789=0.0061 - \mathit{RSS}_1/\mathit{RSS}_0 = 1 - 24.642/24.789 = 0.006.

    The statements above can now be evaluated as right or wrong.


    1. False. An ANOVA always tests the null hypothesis, that all mean values are equal against the alternative hypothesis that they are different.
    2. False. The pp value is 0.9660.966 and hence_not_ significant. It can not be shown that the evaluations differ with respect to the occupation of the respondents.
    3. False. The fraction of explained variance is 0.0060.006 and hence not larger than 0.59.
    4. True. The fraction of explained variance is 0.0060.006 and hence smaller than 0.56.
    5. True. The test statistic is F=0.089F = 0.089 and hence smaller than 1.21.2.