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Univariate Statistics
PSYC*6060
Class 2
Peter Hausdorf
University of Guelph
Agenda
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•
•
•
•
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Review of first class
Howell Chapter 3
Standard distributions exercise
Howell Chapter 4
Block exercise
Hypothesis testing group work
Howell - Chapter 3
• Probabilities
• Standard normal distributions
• Standard scores
Probabilities - Education in Guelph
21%
No High School
29%
24%
14%
12%
High School
Some PostSecondary
Trades/
Certificates
University
Another Example - Diffusion of
% of consumers
Innovation
in each group
adopting the
product
13.5%
34%
34%
16%
2.5%
Early
Innovators Adopters
1997
1999
Early
Majority
2001
Late
Majority
Laggards
Time
Why are distributions useful?
• Understanding the distribution allows us
to interpret results/scores better
• The distribution can help us to predict
outcomes
• Allows us to compare scores on
instruments with different metrics
• Used as a basis for most statistical tests
Standard Normal Distribution
f(X)
.40
0.3413
0.3413
0.1359
0.1359
0.0228
0.0228
0
-2
-1
0
1
2
Standard Scores
• Percentiles
• z scores
P = nL x 100
N
Z=X-X
SD
• T scores
T = (Z x 10)+50
• CEEB scores
A = (Z x 100)+500
Howell - Chapter 4
•
•
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Sampling distribution of the mean
Hypothesis testing
The Null hypothesis
Testing hypotheses with the normal
distribution
• Type I and II errors
Sampling distribution of the
mean
• Standard deviation of distribution
reflects variability in sample statistic
over repeated trials
• Distribution of means of an infinite
number of random samples drawn
under certain specified conditions
Hypothesis testing
•
•
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Establish research hypothesis
Obtain random sample
Establish null hypothesis
Obtain sampling distribution
Calculate probability of mean at least as
large as sample mean
• Make a decision to accept or reject null
The Null Hypothesis
• We can never prove something to be true but
we can prove something to be false
• Provides a good starting point for any
statistical test
• If results don’t allow us to reject the null
hypothesis then we have an inconclusive
result
Testing hypotheses using the
normal distribution
f(X)
:= 25
.40
F= 5
X = 32
0.3413 0.3413
N = 100
0.1359
0.1359
X-:
0.0228
0.0228
Z=
F 0
-2
-1
0
1
2
N
32 - 25
Z = .5
Z = 14, p<.0001, Sig.
Type I error (alpha)
• Is the probability of rejecting the null
hypothesis when it is true
• Border Collies - concluding that they are
smarter than other dogs based on our
study when in reality they are not
• Relates to the rejection region we set
(e.g. 5%, 1%)
Type II error
• Is the probability of failing to reject the
null hypothesis when it is false
• Border Collies - concluding that they are
not smarter than other dogs based on
our study when in reality they are
• Difficult to estimate given that we don’t
know the distribution of data for our
research hypothesis
Relationship between Type I
and Type II Errors
• The relationship is dynamic
• The more stringent our rejection
region the more we minimize Type I
errors but the more we open
ourselves up to Type II errors
• Which error you want to minimize
depends on the situation
Relationship between Type I
and Type II Errors
5% = 1.64
1% = 1.96
f(X) All Dogs
.40
Type I Error
0
-2
-1
0
1
2
Border Collies
Type II Error
-2
-1
0
1
2
Decision Making
True State of the World
Decision
H 0 True
Reject H
Type I error
0 p = alpha
Fail to
reject H
0
Correct decision
p = 1 - alpha
H0 False
Correct decision
p = (1 - beta) = power
Type II error
p = beta
One Versus Two Tailed
Depends on your hypothesis going in. If
you have a direction then can go with one
tailed but if not then go with two tailed.
Either way you have to respect the alpha
level you have set for yourself.