Assignment 5

The number in the the starting parenthesis indicate the corresponding exercise number in “Applied Linear Statistical Models”.

  1. (7 points) (3.12)8 A student does not understand why the sum of squares SSPE is called a pure error sum of squares “since the formula looks like the one for an ordinary sum of squares”. Explain.

  2. (8 points) (Computer project, 3.3) Refer to the GPA data from the previous homework assignments.

    1. Plot residuals e against the fitted values \(\hat Y_i\). What departures from the standard regression assumptions can be detected from this plot?
    2. Prepare a Normal Q-Q plot of the residuals and use it to comment on whether the data passes or fails the assumption of normality. Conduct the Shapiro-Wilk test for normality.
    3. Test whether residuals in this regression analysis have the same variance.
    4. Conduct the lack-of-fit test and state your conclusion.

  3. (8 points) (Computer project) Crime rate data set is available here. A criminologist studies the relationship between level of education and crime rate in medium-sized U.S. counties. She collected data from a random sample of 84 counties; X is the percentage of individuals in the county having at least a high-school dipoma, and Y is the crime rate (crimes reported per 100,000 residents) last year. A linear regression of Y on X is then fit to these data. Test:

    1. normal distribution of residuals;
    2. constant variance of residuals;
    3. presence of outliers;
    4. lack of fit.

  4. (12 points) For the “toy” example, consider a small data set

    X 0 0 1 2
    Y 0 2 2 3

    Try to do as much as you can by hand, without the use of a computer. The numbers are quite simple!

    1. Plot these data and draw the least squares regression line, which has the expression y = 1 + x.
    2. Compute all the residuals.
    3. Compute all sums of squares by hand, from their definitions:

    \[SSTot = \sum_i (Y_i - \bar Y)^2\]

    \[SSReg = \sum_i (\hat y_i - \bar Y)^2\]

    \[SSErr = SSTot - SSReg = \sum_i (Y_i - \hat Y_i)^2\]

    \[SS_{PE} = \sum_j \sum_i (Y_{ij} - \bar Y)^2\]

    \[SS_{LOF} = SSErr - SS_{PE} = \sum_j \sum_i (\bar Y_j - \bar Y)^2\]

    Then conduct the lack-of-fit test. Explain the result.