STAT 516 HW 4 (due Wednesday, March 16)


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 observations per cell).  The weights are 
as follows:

             C
        1(L)     2(S)
       ----    ------
   1    14       14
        10       13
       ----    ------
   2    13       17
A       13       13
       ----    ------
   3    16       18
        14       16
       ----    ------

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 9.23 on page 512 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 are given on the course web page as "roastdata.txt".

Using SAS, 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 P-value for this test.)

(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 
P-value for this test.)

(c) Create an interaction plot (specifically, plot gas against meat 
condition, for each type of oven setup) and state what this 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?  Use an ESTIMATE statement to answer this 
question -- be careful to set it up correctly!  (Give the test statistic 
value and P-value for this test.)

(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? Use an ESTIMATE statement to answer this question -- be 
careful to set it up correctly!  (Give the test statistic value and 
P-value for this test.)

(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.  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 6 observations 
for each combination of levels.  If TSS = 1128, SSA = 375, SSC = 462, and 
SSAC = 38, then:

(a) Write out the COMPLETE ANOVA table (like Table 9.4, page 484) 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.