Review Problems for Midterm Exam 2 - Applied Regression Analysis | STAT 51200, Study notes of Statistics

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Statistics 512: Review Problems for Second Midterm Exam Keep for Second Exam Review (November 17)

1. Short answer questions. Each part is unrelated.

(a) The true means (μ 1 , μ 2 , μ 3 ) in a cell means model have values 40, 50, and 40, respectively. The error variance σ^2 is 20, and the design is balanced with 4 replicates per level. What is the distribution of the following contrast estimate? Lˆ = Y¯ 1 + Y¯ 2 − 2 Y¯ 3 (b) In a 2 × 2 ANOVA, the population means are μ 1 , 1 = 20, μ 2 , 1 = 25, μ 1 , 2 = 31, and μ 2 , 2 = 36. Is there an interaction in this model? Explain why or why not. (c) After performing a least squares regression, you find that the squares of the residuals (r i^2 ) are linearly related to the predictor X according to the following equation: E(r^2 i ) ≈ 1 + Xi. You decide to use weighted linear regression in relating the response Y to the predictor X. Give a formula for the weights wi in terms of the variables in the model. (Assume all the Xi are positive.) (d) Given the following plot of μi versus σi, what transformation of the response would you recommend to make the variance constant?

• •
• •

Group Mean

Group Standard Deviation

5 6 7 8 9

(e) A two-way additive ANOVA model has 3 and 4 levels for the two factors. If the data have 25 observations (i.e., nT = 25), what is the error degrees of freedom? (f) Show that the criterion Cp ≤ p is equivalent to the criterion M SEp M SE(F ull)

≤ 1.

(g) The price per unit increases with lot size until the size is 200. The price per unit stays constant for lot sizes greater than 200. (In other words, the function relating price to lot size is a piecewise linear function. The slope of the line for lot size greater than 200 is zero.)

The output from proc print after reading in the data is

Obs cost lotsize cslope 1 128.7 100 2 107.7 125 3 85.0 150 4 70.3 175 5 48.3 200 6 46.0 225 7 47.6 250 8 47.9 275 9 48.3 300 Fill in an additional column of predictor values (under the cslope heading) that you would use to fit the previously described piecewise linear model. What model statement would you use in proc reg for this model?

2. Refer to the SAS output marked OUTPUT FOR PROBLEM 2.

(a) Write the factor effects model used for this analysis. Include numerical values for the number of levels being compared and the numbers of observations per treatment. Also state the distributional assumption. (b) Summarize the results of the hypothesis tests for main effects and interactions. (c) Explain the results of the Tukey procedures for the main effects. (d) Estimate the residual variance σ^2.

3. Refer to the SAS output marked OUTPUT FOR PROBLEM 3. Greenhouse benches were set up as blocks. Within each block, plants of different genetic varieties were grown. The maximum height of each plant was measured.

(a) Write the cell means model for this analysis. Include numerical values for the number of levels being compared and the numbers of observations per treatment. Also state the distributional assumption. (b) Explain why no interaction term was included in the model. Describe graphical evidence that would justify this assumption (of no interaction)? (c) Write the factor effects model used for this analysis. Then, give a numerical estimate for each parameter in the model.

OUTPUT FOR PROBLEM 2

The GLM Procedure Dependent Variable: Y Sum of Source DF Squares Mean Square F Value Pr > F Model 5 204035.1465 40807.0293 12.91 <. Error 18 56914.1953 3161. Corrected Total 23 260949.

R-Square Coeff Var Root MSE Y Mean 0.781896 18.56378 56.23077 302.

Source DF Type I SS Mean Square F Value Pr > F A 2 25361.0880 12680.5440 4.01 0. B 1 174964.1113 174964.1113 55.34 <. A*B 2 3709.9471 1854.9736 0.59 0.

Source DF Type III SS Mean Square F Value Pr > F A 2 25361.0880 12680.5440 4.01 0. B 1 174964.1113 174964.1113 55.34 <. A*B 2 3709.9471 1854.9735 0.59 0.

Tukey’s Studentized Range (HSD) Test for Y Alpha 0. Error Degrees of Freedom 18 Error Mean Square 3161. Critical Value of Studentized Range 3. Minimum Significant Difference 71. Means with the same letter are not significantly different.

Tukey Grouping Mean N A

A 346.19 8 1 A B A 294.68 8 3 B B 267.85 8 2

OUTPUT FOR PROBLEM 3

The GLM Procedure Dependent Variable: height Sum of Source DF Squares Mean Square F Value Pr > F Model 8 196.5800000 24.5725000 57.95 <. Error 15 6.3600000 0. Corrected Total 23 202.

R-Square Coeff Var Root MSE height Mean 0.968661 3.807911 0.651153 17.

Source DF Type III SS Mean Square F Value Pr > F variety 3 172.9200000 57.6400000 135.94 <. bench 5 23.6600000 4.7320000 11.16 0.

Level of ------------height----------- variety N Mean Std Dev

1 6 18.0000000 1. 2 6 21.0000000 0. 3 6 15.5000000 0. 4 6 13.9000000 1.

Level of ------------height----------- bench N Mean Std Dev 1 4 18.2000000 3. 2 4 16.4000000 2. 3 4 16.3000000 3. 4 4 17.1500000 3. 5 4 18.6000000 3. 6 4 15.9500000 3.