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STAT 350: 97-1

Midterm, 19 February 1997Instructor: Richard Lockhart

Instructions: This is an open book test. You may use notes, text, other books and a calculator. Your presentations of statistical analysis will be marked for clarity of explanation. I expect you to explain what assumptions you are making and to comment if those assumptions seem unreasonable. The exam is out of 25.

  1. Suppose two objects with weights tex2html_wrap_inline96 and tex2html_wrap_inline98 are weighed separately and then together. The resulting data points tex2html_wrap_inline100 , tex2html_wrap_inline102 and tex2html_wrap_inline104 satisfy tex2html_wrap_inline106 , tex2html_wrap_inline108 and tex2html_wrap_inline110 .

    This question is straight from the notes. See Lecture 9

  2. What is the design matrix of this linear model? [2 marks]

    displaymath112

  3. If

    displaymath114

    what is the hat matrix? [2 marks]

    displaymath116

  4. Write tex2html_wrap_inline118 in the form tex2html_wrap_inline120 giving specific numerical values for the tex2html_wrap_inline122 . [2 marks]

    displaymath124

    so that

    displaymath126

    Thus tex2html_wrap_inline128 , tex2html_wrap_inline130 and tex2html_wrap_inline132 .

  5. What is the standard error of tex2html_wrap_inline118 ? [2 marks]

    tex2html_wrap_inline136

  6. What is the variance of the residual corresponding to tex2html_wrap_inline100 ? [2 marks]

    displaymath140

  • A company measures its annual sales Y in each of 26 regions, along with the values of 4 covariates, tex2html_wrap_inline144 , the advertising expenditure in the region, tex2html_wrap_inline146 , the number of active accounts in the region, tex2html_wrap_inline148 , the number of competing brands, and tex2html_wrap_inline150 , a measure of the potential for sales in the region. I attach some SAS code and an edited version of the output.

    1. Is the regression significant? [3 marks]

      displaymath152

      so the regression is definitely significant.

    2. Can advertising expenditure and sales potential be dropped from the full model? [3 marks]

      This is an extra sum of squares F-test. The 2 error sums of squares are 2210.44 and 1937.14 on 23 and 21 degrees of freedom. Thus

      displaymath156

      Compare this to the critical points tex2html_wrap_inline158 and tex2html_wrap_inline160 to see that they may, indeed, be dropped from the model. (The P-value is around 0.25.)

    3. In a model which includes all 4 covariates test the hypothesis that the advertising expenditure is an unimportant predictor. [3 marks]

      The relevant t statistic is 1.67 with a P value of 0.1094 so that we would conclude that advertising expenditure is an unimportant predictor.

    4. What final fitted model seems best? (You will not be able to examine plots or diagnostics). [3 marks]

      The covariates tex2html_wrap_inline146 and tex2html_wrap_inline148 cannot be deleted from the model but tex2html_wrap_inline144 and tex2html_wrap_inline150 can so the best fitted model is

      displaymath176

    5. Give a 95% confidence interval for the coefficient of tex2html_wrap_inline148 . [3 marks]

      You should get the standard error and estimates from the model with tex2html_wrap_inline146 and tex2html_wrap_inline148 . This gives

      displaymath184

    Complete SAS output.

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    Richard Lockhart
    Wed Feb 19 14:43:05 PST 1997