Condence Intervals
A condence interval provides a simple summary of
how precisely a parameter, denoted , is estimated.
In many situations, a (1 )100% condence interval is of the form
( s t/2 ,
+ s t/2 )
where
is an estimate of
s is its standard error
Hypothesis Testing
basic ingredients of a hypothesis test are
1. the null hypothesis, denoted Ho
2. the alternative hypothesis, denoted Ha
3. the test statistic
4. the the data
5. the conclusion
the hypotheses are usually statements about the values of
1
Wilcoxon Rank-Sum Test, also known as the Mann-Whitney test
Rank the data. That is replace, the data values by their ranks, from
smallest to largest. For example, the pH samples are:
Group 1:
Group 2:
8.53 8.52 8.01 7.99 7.93
7.85 7.73 7.58 7.40 7.35
7
Comparing two means, paired experiment
many studies are comparative
they compare outcomes from one group with outcomes from another
(e.g. two dierent medical treatments)
in the matched-pairs design each subject in one group is paired with a similar su
Introduction, 1 sample t-test and t-interval
Central Limit Theorem
Let X1 , X2 , . . . , Xn be a random sample from a distribution with mean
and variance 2 .Then if n is suciently large (Rule of thumb > 30), X has
2
2
approximately a normal distribution
Permutation Test
for the Two Sample Problem
we wish to compare results for two groups
of experimental units
the rst group could be some subjects
who have been given a treatment, whereas
the second group has not
in some cases we are unable to assume
tha
1
One-Way Analysis of Variance (ANOVA)
One-Way Analysis of Variance (ANOVA) is a method for comparing the means of a
populations. This kind of problem arises in two dierent settings
1. When a independent random samples are drawn from a populations.
2. Whe
1
Kruskal-Wallis test (a non-parametric analogue of one-way ANOVA)
The assumptions of the usual one way ANOVA are:
Xij = i + eij
where eij are independent N (0, 2 ), i = 1, 2, . . . , a, j = 1, 2, . . . , ni
If a normal scores or normal probability plot o
Arthur Schnitzler
RHAPSODY
A DREAM NOVEL
Translated from the German by
OTTO P. SCHINNERER
RHAPSODY
1
"TWENTY-FOUR brown-skinned slaves rowed the splendid galley which was to bring Prince Amgiad to
the palace of the caliph. The Prince, wrapped in his purpl
Stat 2080 Assigment 9 Solutions Winter 2010
1. An exercise physiologist used skinfold measurements to estimate the total body fat, Y ,
expressed as a percentage of body weight, X1 , for 19 participants in a physical fitness
program. The body fat percentag
Stat 2080 Assigment 5 Solutions Winter2010
1. Question 1
In a study to compare the visibility of paints used on highways, four different types of
paint were tested at three different locations. It is to be expected that paint wear will
differ depending on
Stat 2080 Assigment 6 Solutions
1. In Assignment 5 you analyzed data on the number of spruce moths found in traps
after 48 hours. The traps were placed in different locations in the tree (top, middle,
lower branches, and ground) and different types of lur
Stat 2080 Assigment 7 2010 Solutions
1. A random sample of individuals who drive alone to work in a large metropolitan area was obtained,
and each individual was categorized with respect to both size of car and commuting distance.
Does the accompanying da
Stat 2080 Assignment 8 Solutions 2010
1.5
1. For this question refer to the plot shown on OWL labeled Figure for Assignment 8 in the file
Assignment Images and Resources. The plot contains 4 data points (shown as circles) and 3 lines
(labeled A, B, and C)
Randomized Block Example
A farmer wishes to compare the growth times
of four dierent varieties of daodil under a range
of dierent conditions. She decides to use a randomized block design where six elds are split
into four equally sized plots and then the
1
Two-Way Analysis of Variance - no interaction
Example: Tests were conducted to assess the eects of two factors, engine type, and propellant type, on
propellant burn rate in red missiles. Three engine types and four propellant types were tested.
Twenty-f
Subsequent Inferences for one-way ANOVA
if the overall F test does not show
signicant dierences among the groups, no further inferences are required
if the overall test does show a signicant dierence, dierences between particular means can be tested usi
1
Matrix Methods for Simple Linear Regression
1. As before, the two normal equations must be solved simultaneously to obtain estimates
of regression coecients, 0 and 1 .
2. Here we start by factoring the two normal equations into matrix form.
n
n
n0 + 1
x
Introduction, 1 sample t-test and t-interval
Central Limit Theorem
Let X1 , X2 , . . . , Xn be a random sample from a distribution with mean
has
and variance 2 .Then if n is suciently large (Rule of thumb > 30), X
2
2
approximately a normal distribution
Confidence Intervals
A confidence interval provides a simple summary of
how precisely a parameter, denoted , is estimated.
In many situations, a (1 )100% confidence interval is of the form
( st/2 ,
+ st/2 )
where
is an estimate of
s is its standard e
Data generated from a normal distribu+ons .
-0.5
-1.0
-1.5
-2.0
y
0.0
0.5
Pearson Corr =
0.84
Spearman Corr=
0.78
-2.0
-1.5
-1.0
-0.5
0.0
0.5
x
Pearson correla+on (usual) one and Spearman
correla+on (non-parameteric one) give similar v
Two ways to look at Con-ngency Tables
Species distribu-ons by Sample
Sample distribu-ons by Species
Minitab: Stat -> Tables -> Chi-Square Test (Two-Way Table in Worksheet)
Test Sta-s-c is 4.601, DOF = 6, P-value=