# Mathematics

SRM 502 Homework Assignment 4

Answer the following questions. For “hand calculations” feel free to use anything like Excel to make the calculations easier and quicker. Use SAS only when directed.

1. A researcher studied the sodium content in lager beer by selecting at random six brands from the larger number of brands of U.S. and Canadian beers sold in a metropolitan area. She then chose eight 12-ounce cans or bottles of each selected brand at random from retail outlets in the area, and measured the sodium content ( in milligrams) of each can or bottle. The observation follow

observation Brand 1 2 3 4 5 6 7 8

1 24.4 22.6 23.8 22.0 24.5 22.3 25.0 24.5 2 10.2 12.1 10.3 10.2 9.9 11.2 12.0 9.5 3 19.2 19.4 19.8 19.0 19.6 18.3 20.0 19.4 4 17.4 18.1 16.7 18.3 17.6 17.5 18.0 16.4 5 13.4 15.0 14.1 13.1 14.9 15.0 13.4 14.8 6 21.3 20.2 20.7 20.8 20.1 18.8 21.1 20.3

(a) What type of model would be most appropriate to answer the researcher’s question?Why?

(b) Write an appropriate model, according to your answer for part a.

(c) Write appropriate null and alternative hypotheses that can be used to answer the re- searcher’s question.

(d) Using SAS, perform a test of these hypotheses. Explain the conclusions that can be made in terms of the language of the problem. Use α = 0.05.

(e) Estimate the variance σ2a.

2. In three species of citrus trees the ratio of leaf area to dry weight was determined for three conditions of shading.

species Shading Shamouti Orange Marsh Grapefruit Clementine Mandarin

Sun 112 90 123 Half shade 86 73 89

Shade 80 62 81

A researcher is interested in effectiveness of shading in decreasing the relative leaf area.

(a) What type of model would be most appropriate to answer the researcher’s question?Why?

(b) Write an appropriate model, according to your answer for part a.

(c) Write appropriate null and alternative hypotheses that can be used to answer the re- searcher’s question.

(d) Calculate the sum of squares associated with each factor by hand, and use these to con- struct an ANOVA table according to this decomposition of the model sum of squares. Show your work.

(e) Perform the F-test to answer the researcher’s question, and explain the result. Use α = 0.05.

3. A recent study was conducted to assess the effectiveness of a new weight-loss drug. After random selection, eight mice were given a single dosage, eight mice were given a double dosage, and eight mice were given a placebo. Drug 1 is a single dosage and Drug 2 is a double dosage. The gender of the mice was also recorded, with four males and four females in each group. The data follow.

Drug 1 2 Placebo Gender 7 21 5 M 7 14 5 M 9 17 9 M 6 12 7 M

10 16 7 F 8 14 6 F 7 14 9 F 6 10 8 F

(a) Write an appropriate Two-Factor ANOVA model.

(b) Using SAS, obtain interaction plots for the Drug*Gender interaction effect.

(c) Use SAS to evaluate the significance of the Drug*Gender interaction effect, with α = 0.05.

(d) Use SAS to evaluate the significance of the Drug main effect, and also for the Gender main effect, with α = 0.05.

(e) Explain the results of parts c and d in terms of the language of the problem.

4. A researcher has obtained data from 45 subjects to compare three methods of therapy: rational-emotive therapy (RET), client-centered therapy (CCT), and behavior modification (BMOD). Three therapists were employed: each therapist treated five clients with each method of therapy. Assume the following ratings of the effectiveness of the therapy were obtained.

Therapist 1 2 3

Method RET 40, 42, 36, 35, 37 40, 44, 46, 41, 39 36, 40, 41, 38, 45 CCT 42, 39, 38, 44, 42 41, 45, 40, 48, 46 41, 39, 37, 44, 44

BMOD 48, 44, 43, 48, 47 41, 40, 48, 47, 44 39, 44, 40, 40, 43

(a) What type of model would be most appropriate to answer the researcher’s question?Why?

(b) Write an appropriate model, according to your answer for part a.

(c) Write appropriate null and alternative hypotheses that can be used to answer the re- searcher’s question.

(d) Using SAS, perform a test of these hypotheses. α = 0.05.

(e) Explain the conclusions that can be made in terms of the language of the problem.

5. A research was conducted to investigate the capability of measurements in thermal impedance ( C◦/w × 100) on a power module for an induction motor starter. There are 10 parts, three inspectors, and three replicates. The data are shown below.

Inspector 1 Inspector 2 Inspector 3

Test Test Test

Part No. 1 2 3 1 2 3 1 2 3

1 37 38 37 41 41 40 41 42 41 2 42 41 43 42 42 42 43 42 43 3 30 31 31 31 31 31 29 30 28 4 42 43 42 43 43 43 42 42 42 5 28 30 29 29 30 29 31 29 29 6 42 42 43 45 45 45 44 46 45 7 25 26 27 28 28 30 29 27 27 8 40 40 40 43 42 42 43 43 41 9 25 25 25 27 29 28 26 26 26 10 35 34 34 35 35 34 35 34 35

(a) Explain why we can consider both parts and inspectors as random factors?

(b) Assuming that both parts and inspectors are random, write the appropriate model equa- tion and its assumptions.

(c) Use SAS to analyze the data, assuming that both parts and inspectors are random. For each test, write the appropriate null hypothesis.

(d) Use SAS to estimate the variance components using the ANOVA method.