Chapter 2: Two-Sample Methods
8 Selecting Among Two-Sample Tests Assumptions X1 , . . . , Xm iid F1 and Y1 , . . . , Yn iid F2 (F1 and F2 are continuous cumulative distribution) X1 , . . . , Xm and Y1 , . . . , Yn are independent : the cdfs dier by F1 (x)
STA5507 Section 8112 EXAM 1. Solutions
1. Below are recorded the white blood cell counts for a sample of 6 adult males with leukemia (recorded in thousands of cells). 19.3 32.0 1.8 43.0 14.1 1.4
Making no assumptions about the distribution of white cell c
STA4502 Section 2906 EXAM 2 Solutions
1. Comparison of peak expiratory ow rate (PEFR) before and after a walk on a cold winters day for a random sample of 9 asthmatics. Subject Before After 1 312 300 2 242 201 3 340 232 4 388 312 5 296 220 6 254 256 7 391
STA5507 Section 8112 EXAM 2 Solutions
1. Comparison of peak expiratory ow rate (PEFR) before and after a walk on a cold winters day for a random sample of 9 asthmatics. Subject Before After 1 312 300 2 242 201 3 340 232 4 388 312 5 296 220 6 254 256 7 391
STA4502/5507 EXAM 3 Solutions
Name: UFID#:
1. Featherston (1971) was interested in the relationship between the weight of tapeworms fed to dogs and the weight of the scoleces recovered from the dogs after 20 days. The cysticerci used in the experiment wer
STA4502/STA5507 Extra Credit Homework Solutions
1. Johnson (1984) sampled the young-of-year (YOY) gizzard shad (Dorosoma cepedianum) population at four dierent sites in Kokosing Lake (Ohio) in summer 1984. The data are lengths (mm) of the YOY gizzard shad
Homework 1. (pp.21-22) Exercise #1,2,3,4, and 5 Extra credit problem pp.22 Exercise #7 Homework 2. (pp.73-75) Exercise #2.b, 3, 4 ( for a. using a permutation test of the dierence of means), 5, 7, 8, 10, 12, 13, 15 ( for a. using the Ansary-Bradley test
Homework 1 Solutions
1. The Binomial Test
H0 : 0.5 = 70 Ha : 0.5 > 70,
Under H0 ,
P r (X 70) = P r (X > 70) = 0.5.
Thus, hypothesis can be re-written as
H0 : p = 0.5 Ha : p > 0.5.
Test Statistics:
n
I (xi > 70) = 38
b=
i=1
Normal Approximation:
B 0.5 n
z
Homework 2 Solutions
2.b. & 3. Permutation distribution of the dierences of means and medians
H0 : F1 (x) = F2 (x)
Ha : F1 (x) < F2 (x) or F1 (x) > F2 (x)
For the p-value calculation,
p valuemean di
N
1
=
N
I (|Di | |Dobs |)
i=1
p valuesum trt1 = 2
p val
Homework 3 Solutions
1. & 3. K -sample Permutation F -test and one-way ANOVA/ Kruskal-Wallis test H0 : 1 = 2 = 3 Ha : at least one of i = j
_ ANOVA Table: DF SS MS F p-value Treatment 2 7.14 3.57 2.99 0.088 Error 12 14.32 1.19 NA NA Total 14 21.45 NA NA N
Homework 4 Solutions
1. & 2. The eosinophil counts from 40 healthy rabbits bootstrap estimates -Basic Statistics from the original data Mean= 124.78 SD= 58.45 CV= 46.84 -Number of bootstrap saples: 5000 Mean SD CV MSE 82.196 67.739 27.062 Standard Error 9
Homework 5 Solutions
1. & 3. The permutation distribution of the mean of the dierences and of the signedrank statistic
-Number of All Possible Permuations: 16
-Permutation distributions of the mean of the differences
d.bar
-8
-7
-4
-3
-2
-1
0.0625 0.0625
Homework 6 Solutions
1. & 2. The permutation distribution of the slope, Spearmans rs and Kendalls for
the height and weight
-Number of All Possible Permuations: 6
-Permutation distribution of the slope of the least squares line
-2.5 -1.96 -0.36 1.25 2.32
Chapter 0. Preliminaries
1. Some Denitions A Population is the entire collection of objects or outcomes about which information is sought. A Sample is a subset of a population, containing the objects or outcomes that are actually observed. A Simple Random
Chapter 1: One-Sample Methods
1. Non-Parametric Test of Hypothesis and CI for the Median
Binomial Test (Sign Test)
X1 , . . . , Xn iid F (x) where F (x) is a continuous cdf
0.5 : the population median
Test
H0 : 0.5 = 0
Ha : 0.5 > 0
Dene
n
B=
i where
Chapter 2: Two-Sample Methods
1. Two-Sample Permutation Test Two-sample t-test Assumptions 2 2 X1 , . . . , Xm iid N (X , X ) and Y1 , . . . , Yn iid N (Y , Y ) X1 , . . . , Xm and Y1 , . . . , Yn are independent 2 2 X = Y Test H0 : X = Y vs Ha : X > Y Po
Chapter 2: Two-Sample Methods
3 Wilcoxon Rank-Sum Test
Assumptions:
X1 , . . . , Xm iid F1 and Y1 , . . . , Yn iid F2
(F1 and F2 are continuous cumulative distribution)
X1 , . . . , Xm and Y1 , . . . , Yn are independent
No tie in observations
Rank o
STA4502 Section 2906 EXAM 1. Solutions
1. Below are recorded the white blood cell counts for a sample of 6 adult males with leukemia (recorded in thousands of cells). 19.3 32.0 1.8 43.0 14.1 1.4
Making no assumptions about the distribution of white cell c
Tukeys HSD Critical Values: q(alpha, k, df)
*The c ritical values f or q c orres ponding to alpha = .05 (top) and alpha = .01 (bottom) df 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 24 30 40 60 120 in f i ni ty k= Number o f Treatments 2 3 4 5 3.64 5.70 3.
Chapter 3: K -Sample Methods
0. Comparing more than two groups (treatments)
overall comparison: whether or not dierences exist among groups
multiple comparison: which groups dier signicantly from the others
Assumption: the experimental units are assign
Chapter 3: K -Sample Methods
3 Multiple Comparisons Which treatments dier from the others among more than two treatments?
pairwise test: by doing so many pairwise comparisons, the probability of declaring at least two treatments to be dierent may be cons
Chapter 8: Nonparametric Bootstrap Methods
1. The Basic Bootstrap Method
to measure how close the statistical estimate is to the population parameter
for the population mean
the approximate 95% CI is given by
SX
X 2 .
n
the margin of error 2SX n1/2 :
Chapter 8: Nonparametric Bootstrap Methods
4. Two-Sample Inference the problem of making a condence interval for the dierence between the means of two populations using the bootstrap method Notation Yij = i + ij Fi (y ) j = 1, 2, . . . , ni i = 1, 2
t-Pi
Chapter 4: Paired comparisons and Blocked Designs
Paring and Blocking are experimental design techniques that enable a researcher to detect dierences among treatments more easily in environments that have a lot of variability among experimental units. 1.
Chapter 4: Paired comparisons and Blocked Designs
4. A Permutation Test for a Randomized Complete Block Design Randomized complete block design (RCBD) blocking: when the experimental units to which treatments are to be are not homogeneous, or when the con
Chapter 5: Tests for Trends and Association
the relationship between two quantitative variables - correlation and regression/
nonparametric correlation and regression
the relationship between qualitative variables - contingency tables
1. A Permutation T
Chapter 5: Tests for Trends and Association
4. Permutation Tests for Contingency Tables
A two-way contingency table: the counts of individuals who fall into each of
the categories that are determined by the two characteristics being studied.
Data layout
Chapter 5: Tests for Trends and Association
6. Contingency Tables with Ordered Categories alternatives to the 2 statistic when there is an ordering among the categories of one or both factors in a contingency table Singly Ordered Tables the categories of
Chapter 8: Nonparametric Bootstrap Methods
4. Correlation and Regression
bootstrap approach to making inferences about the correlation coecient
and the slope of a regression line
Bivariate Bootstrap Sampling
random sample (Xi , Yi ) i = 1, . . . , n
b