STATISTICS 402 - Assignment 5 (Due March 25, 2005)
This activity involves four treatments; A, B, C, and D and five blocks; white, blue, green, yellow
and pink. The treatments could be food additives, the blocks litters of mice, and the response
amount of weight gained. Alternatively, the treatments could be a combination of starting technique (without starting blocks, with starting blocks) and shoes (no spikes, spikes) with the response
the time to run a 400 meter race, blocks would be different individuals.
The activity consists of two different randomizations, complete randomization and restricted randomization (blocking) and their subsequent analyses.
1. Complete Randomization
• Put all 20 squares in a bag.
• Shake well. Draw out a square, Record the value for treatment A. DO NOT RETURN
the square to the bag. Repeat this process: Shake, Draw, Record until you have 5 values
for treatment A.
• Repeat the step above for treatments B and C.
• There should be 5 squares remaining in the bag, use these for treatment D.
2. Analysis
• Compute treatment means and construct an Analysis of Variance table.
• Report the value of the F-test and P-value for comparing treatments. What do these
tell you about the differences between the various treatments?
• Note particularly the value for the mean square error.
• If the F-test indicates that there are some statistically significant differences follow up
with a multiple comparison procedure. That is, compute a value for the adjusted Least
Significant Difference and use it to determine which, if any, of the treatment means differ
significantly.
v
u
u
∗t
adjLSD = t M SError
1
1
+
ni nj
!
• Look at plots of residuals versus treatments and a normal plot of residuals. What do
these tell you about the conditions necessary for the analysis above? Be sure to clearly
indicate which plot tells you about which condition.
3. Restricted Randomization, Blocking
• Put the 4 white squares in a bag.
• Shake well. Draw out a square, Record the value for treatment A. DO NOT RETURN
the square to the bag. Shake, Draw, Record the value for treatment B. Shake, Draw,
Record the value for treatment C. Take out the last square and Record the value for
treatment D.
• Repeat the above steps for the blue, green, yellow and pink squares.
1
4. Analysis
• Compute treatment means and construct the appropriate analysis of variance table.
• Report the value of the F-test and P-value for comparing treatments. What do these
tell you about the difference between the various treatments?
• Note particularly the value for the mean square ERROR.
• If the F-test indicates that there are some statistically significant differences follow up
with a multiple comparison procedure. That is, compute a value for the adjusted Least
Significant Difference (Bonferroni) and use it to determine which, if any, of the treatment
means differ significantly.
v
u
u
∗t
adjLSD = t M SERROR
1
1
+
ni nj
!
• Look at plots of residuals versus treatments and a normal plot of residuals. What do
these tell you about the assumptions underlying the analysis above?
5. Comparison
• Compare the results of the two analyses. With each design/analysis indicate which pairs
of treatment means have statistically significant differences.
• Compare the estimates of error variability for the two designs, complete randomization
and restricted randomization. What do you notice?
• What does this say about the use of restricted randomization (blocking)?
• Make a plot of the treatment values for each block. Does this plot indicate any potential
interaction between treatments and blocks? If it does, what problems might this cause
in the analyses above?
2