STAT 516 HW 6


HAND CALCULATIONS:

1.  We will analyze the data in the following table.  The response variable is weight of 
a tomato plant in pounds, and Y values are given in the table for all combinations of the 
factor levels.  The factors are Fertilizer (labeled factor A) and Light Source 
(labeled factor C).  We have 3 types of fertilizer and 2 types of light source (lamp and 
sunlight).  Each combination of factor levels had two plants assigned to it (i.e., there 
were two replications).  The weights are as follows:

             C
        1(L)     2(S)
       ----    ------
   1    14       15
        10       11
       ----    ------
   2    13       15
A       10       13
       ----    ------
   3    16       23
        14       19
       ----    ------

Do the following by hand, SHOWING WORK.  You may check your answers with SAS if you want.

(a) Find the overall mean of the 12 plants (this is y-bar-dot-dot-dot).  Also find the means 
for the six cells (these are the y-bar-i-j-dot values).

(b) Subtract the overall mean from each Y value.  Square each of the results, and add 
those squares up.  This is TSS.

(c) Subtract the corresponding cell mean from each Y value. Square each of the results,
 and add those squares up.  This is SSW.

(d) Find SS(Cells) by subtracting SSW from TSS.

(e) Find MSW and MS(Cells) by dividing by the appropriate degrees of freedom in this 
example.

(f) Find the F-ratio, and determine (at a 5% significance level) whether the population 
cell means are equal by comparing the F-ratio to the appropriate F value.

COMPUTER CALCULATIONS:

2.  Table 9.24 on page 456 lists the gas requirements for cooking a roast to 160 degrees 
under several conditions.  The factors are Oven Setup (levels:  "Preheated" or "Not 
Preheated") and Meat Condition (levels:  "Fresh", "Frozen", "12-hour Thawed", and "24-hour 
Thawed").  There were 5 observations assigned to each combination of factor levels.  The 
data and SAS code to read it are given on the course web page as "roastdata.txt" 
(or "roastdataR.txt" for R users).

Using SAS or R, answer the following questions:

(a) At a 5% significance level, test whether we can conclude that there is a significant 
difference in mean gas usage across cells.  Give the test statistic value and the P-value.

(b) At a 5% significance level, test whether there is a significant interaction between 
Oven Setup and Meat Condition.  If so, interpret what this result is saying.  Give the 
test statistic value and the P-value of the test.

(c) Create an interaction plot (specifically, plot gas against meat condition, for each type 
of oven setup) and state what they tell you about the joint effect of Oven Setup and Meat 
Condition on gas usage.

(d) At a 5% significance level, is there a significant difference in mean gas usage between 
preheated and non-preheated ovens, when the meat is thawed for 24 hours?  Write the form of the 
contrast you would use to test this formally.  Look at the SAS output "roastdataoutput" on the
course web page.  Based on this output, answer the question in this part. 
Give the test statistic value and the P-value.

(e) At a 5% significance level, is there a significant difference in mean gas usage 
between fresh meat and all non-fresh meat, when the oven is preheated?  Write the form of the 
contrast you would use to test this formally.  Look at the SAS output "roastdataoutput2" on the
course web page.  Based on this output, answer the question in this part.  
Give the test statistic value and the P-value.  

(f) Consider Tukey's procedure for checking for significant differences between each pair of 
factor levels.  At the 5% level, two combinations have a significantly different mean gas 
usage if their sample means differ by more than what value? 
(Hint: the critical value from Table A.7 in this case is q_.05(t=8,df=32).)

CONCEPT QUESTIONS

3. When we have no replication (i.e., one observation for each combination of factor 
levels), why must we typically assume that the most complicated interaction does not exist?

4.  Suppose we have a factorial experiment with two factors, A and C.  Factor A has 5 levels 
and Factor C has 3 levels.  There are 4 observations for each combination of levels.  If 
TSS = 1227, SSA = 277, SSC = 412, and SSAC = 113, then:

(a) Write out the COMPLETE ANOVA table (like Table 9.4, page 427) for the experiment.  

(b) At a 5% significance level, is there a significant interaction between A and C?
Provide numerical evidence for your answer.

(c) At a 5% significance level, is there a significant difference in mean response 
across the levels of factor A?  Provide numerical evidence for your answer.

(d) At a 5% significance level, is there a significant difference in mean response 
across the levels of factor C?  Provide numerical evidence for your answer.