For the theory section show your work and do not round off your middle work Keep your final answer to four decimal places where relevant To compare whether the mean level of sat

For the theory section, show your work and do not round off your middle work.  

Keep your final answer to four (4) decimal places, where relevant.

1.         To compare whether the mean level of saturated fat is different in three different brands of margarine (A, B and C), a researcher took five (5) samples each of the three different brands and analysed the level of fat. The result given below is the fat content (in grams) in a 20-gram serving:

Brand A            Brand B            Brand C

12.5      15.3      14.2

13.2      15.4      13.9

12.8      14.6      14.3

13.0      15.0      14.5

12.9      14.5      13.8

a.          Use Hartley’s F test and verify whether the condition about variance is satisfied.

b.         Can it be concluded at the 5% significance level that there are differences in the mean level of saturated fat between the three brands. [Assume that all requirements are satisfied for this test.]

c.          What happens to the F statistic when the sample size increases; for example, from 5 to 15. Justify your answer. 

2.         In an experiment with t = 5; and n1 = 5; n2 = 7; n3 = 3; n4 = 8; n5 = 7 — suppose we define the following contrasts: 

        C1 = y1 + y2+ 2y3 - 3y4 - y5

        C2 = y1 + y2  - 2y4

 a.         Are these contrasts linear? Justify your answer.

b.         Are these contrasts orthogonal? Justify your answer.

3.         What is a Type I error?  Distinguish between experiment-wise Type I error rate and comparison-wise Type I error rate.   Which of these error rates is often controlled by researchers via use of methods such as the Bonferonni adjustment and why?  

4.         In a research study involving eight (8) treatments, five contrasts were selected by the researcher for comparing various hypotheses about the treatment means.

a.          If we want to control the experiment-wise error rate at a level of alphaE = 0.05, using the Bonferroni inequality, what value should be selected for alphaI.

b.         What is the critical value from the F table for testing the five contrasts if the sample size is 6 for each of the 8 samples. 

5.         A study was conducted to compare five different types of candies. The variable of interest was the “taste value.” A group of 300 people were asked to rate the taste from 1 (poor) to 4 (excellent), after tasting the candies. The candy types were

Type A: Chewy from Expensive Brand 1

Type B: Crunchy from Expensive Brand 1

Type C: Chewy from Inexpensive Brand 2

Type D: Crunchy from Inexpensive Brand 2

Type E: Crunchy from Expensive Brand 3

The mean ratings for the five types are given below:

Type     A          B          C          D          E

Mean Rating      2.32      2.56      3.22      2.94      3.76

The mean square error is 0.2457.

a.          Consider a contrast of {-2, -2, 3, 3, -2}. Please explain in words which combination of the 5 types of candy in this study is being compared with which other combination of the 5 types of candy in this study by this contrast.  (Hint:  Striving to understand the 4 contrasts and their associated a1, a2, a3, a4, and a5 values shown in Table 9.3 on page 111 in the textbook might prove helpful to you with answering this question.)

b.         What is this value of the contrast?

c.          What is the sum of squares for this contrast?

d.         Is there significant evidence that the mean value of the contrast is not zero?

6.         a.          Use Fisher’s LSD method with alpha = 0.05 to determine which population means differ in the following problem:

t = 3; MSE = 800

Sample size       20         20         20

Sample mean

130       100       140

b.         Repeat the experiment using the Bonferroni adjustment.

c.          Use Tukey’s multiple comparison method to see which population means differ in the given problem.

7.         Consider the following treatment means from an experiment with seven (7) observations per treatment. The MS error = 0.8. Use Scheffe’s method for testing the significance of the contrasts with coefficients {-3, -1, +2, +1, +1}. Use 0.05 as the level of significance.

Treatment          A          B          C          D          E

Mean    3.0        2.0        2.5        1.5        4.5

8.         An experiment was conducted with 5 treatments and 4 observations per treatment. Treatment A was the control. The treatment means and the means square error are shown below. Which if the other treatments are significantly less that the control at the 5% level of significance. 

Treatment          A          B          C          D          E

Mean    70         80         60         90         85

 MSE = 20.