ST 524 NCSU - Fall 2008
Homework 2 Due: Thursday September 25, 2008
I. An evaluation of sample size is carried on paper “Sample Size for Single Double and
Three-Way Hybrid Corn Ear Traits”
1. Statistical Design
2. Response variables
Write down the statistical linear model for yield of grains at 13% humidity, in words if preferred.
3. The Analysis of variance table for Yield is presented next,
How many blocks should be used if the minimum detectable difference ( in any direction to be found between any two hybrids is 5 ton h -1 at a significance level of 0.05 and power equal 0.9
 )
Use r

2
 z

2
 z


2


2
,
4. Subsampling
Tuesday September 15, 2008 1

ST 524 NCSU - Fall 2008
Write down the statistical linear model for response Weight of 100 grains.
5. Analysis of variance table for Weight of 100 grains is presented next, a) Explain differences between Experimental and Sampling Error. b) Experimental Coefficient of variation (Exp. CV) is calculated by
24.2%

61.5

100
, while Am CV corresponds to
13.6%
32.4
How would you interpret these two values?

19.6
32.4

100
. c) Calculate the variance components in the Expected Experimental Error MS. d) Researchers want to analyze the increase on the number of blocks and/or number of ears per plot, necessary to find significance for the set of effects similar to the observed in table 3. What combinations of r and s would you suggest? Consider power = 0.80
Weight of 100 grains (g)
Type Hybrid Mean Variance CV
Single
Single
Three-way
Three-way
Double
Double
P30F33
P Flex
AG 8021
DG 501
AG 2060
DKB 701
33.2
33.5
33.8
31.1
30.5
32.3
35.10
12.47
17.57
14.10
16.83
21.43
17.8
10.5
12.4
12.1
13.4
14.3
Standard
Deviation
5.9245
3.5313
4.1917
3.7550
4.1024
4.6293
Overall 32.4 19.6 13.6 4.4272 e) Calculate the Standard Error of a hybrid mean for each case, where


2

 rs r
2 e . Which combination of r and s in d) will result in the smallest standard error for the mean?
Tuesday September 15, 2008 2

ST 524 NCSU - Fall 2008 f) The optimum allocation solution for number of ears selected within each
2 plot, when minimizing the total cost, is given by s
 , where s is the c s
2 e
2 number of ears selected within each plot, c
1
is the management cost for each plot and c
2
is the cost per ear sampled, find the optimum allocation number of ears per plot for a relative cost c and s e
2   ˆ 2 e

8.38
1
= 3.0 and c
2
= 0.1, with s
2   ˆ 2 
19.6
. Do any of the suggested combinations of (r, s) for the desired power correspond with the optimum number of ears to be sampled within each plot? g) Through independent calculations of sample size, researchers found that three blocks and 21 ears per block were adequate to attain a CV = 5%, an improvement over the observed CV average (Am CV) of 13.6%.
Comment the following
Use the standard error of the mean to compare a design with three blocks and 21 ears subsampled within each plot, with a design with 7 blocks and 9 ears per plot. What power is associated with each scenario? h) Calculate the sample size needed to estimate the observed difference between
Single and Double Hybrids with a power = 0.80. i) Calculate the sample size needed to estimate the observed difference between
Single and Three-way Hybrids with a power = 0.80. j) Does any of the suggested sample size agree with the needed number of blocks found in 3?
Tuesday September 15, 2008 3