Transcript Describing a sample

EDUC 200C
week10
December 7, 2012
Two main ideas…
• Describing a sample
– Individual variables (mean and spread of data)
– Relationships between two variables (correlation)
• Making inferences about the population from
the sample
– One sample (t-test)
– Two samples (t-test)
– Two or more samples (ANOVA)
DESCRIBING A SAMPLE
Describing a sample
• Individual variables
– Central tendency
• Mean, median, mode
– Variability
• Spread of observations around the mean
• Variance
• Standard deviation
Describing a sample
• Relative position
– z scores
– Data transformation to give data a mean on 0 and a
standard deviation of 1
Describing a sample
• The relationship between two ore more variables
– Measure of the strength of relationship
– Pearson correlation (between two continuous variables)
• Z-score difference formula
• Z-score product formula
• Raw score formula
– Spearman rank-order correlation coefficient (two rank
order variables)
Describing a sample
• Regression
– Predict Y from X:
–
–
– Error (or residual):
– Standard error:
– r-squared:
INFERENCE
The Normal Distribution
Inference
• Type I and Type II error
H0 True
H0 False
Reject H0
Type I error
α
Correct!
Power: 1-β
Retain H0
Correct!
Confidence: 1-α
Type II error
β
Inference
• Power reflects our ability to correctly reject the null
hypothesis when it is false
• Must have a specific alternative hypothesis in mind
– Alternatively, we can specify a target power level and,
with a particular sample size determine how big of an
effect we will be able to detect
• We have higher power with larger samples and
when testing for large effect sizes
• There is a tradeoff between α and power
Inference
• One Sample
– H0: μ=some number
– Population standard deviation (σ) known
• Standard error:
•
• Compare to normal distribution
• Confidence interval:
– Population standard deviation not known
• Standard error:
•
• Compare to t distribution
• Confidence interval:
Inference
• Two samples
– Independent samples
• H0: μ1= μ2
• Pooled variance:
• Standard error:
•
• Confidence interval:
Inference
• Matched pairs
–
– H0: μD=0
– Standard error:
–
– Compare to t distribution
Inference
• More than two samples
–
– Compare to F distribution
– One-way ANOVA
• H0: μ1= μ2 =…= μk
– Two-way ANOVA (factorial design)
• H0: μa1= μa2 =…= μaj
μb1= μb2 =…= μbl
μaxb1= μaxb2 =…= μaxbk
– Degrees of freedom will vary with number of groups
and levels within factors
Concept Map: Descriptive
Descriptive
Statistics
One
Variable
Central
Tendency
Mode
X
X 
N
The most
frequent Xi
The value of the
middle case
(if N is odd)
Relative
Standing
Variability
Median
Mean
Two
Variables
s2 
Variance
 (X  X )
Standard
Deviation
2
N 1
The average of the
values of the two
middle cases
(if N is even)
s
(X  X )
N 1
Percentile
Rank
Z score
2
z
X 

X X
z
s
Frequency
Correlation
Pearson
Correlation
Coefficient
(interval/ratio)
or
r

Z X ZY
N 1
Regression
Spearman
Correlation
Coefficient
(Ordinal)
rs  1 
6

D2
N ( N 2  1)
Slope
Intercept
sY
sX
a  Y  bX
br
Concept Map: Inferential
Inferential
Statistics
Compare
Means
One
Group
Two
Groups
Population
SD Known
Population
SD Unknown
Two
Independent
Groups
z-test H0:
µ = Constant
t-test H0:
µ = Constant
t-test H0:
µ1 = µ 2
z
X 
/ N
t
Examine
Associations
X 
s/ N
df  N  1
t
More than
Two Group
Two Paired
Groups
One-way
ANOVA H0:
µ1 = µ 2 = µ p = µ
t-test H0:
µD = 0
X1  X 2
 ( N1  1) s12  ( N 2  1) s22   1
1 


*

N1  N 2  2

  N1 N 2 
df  N1  N 2  2
One
Factor
t
D
sD / N
df  N  1
F
MS Between
MSW ithin
df B  g  1
dfW  N  g
Correlation
between Two
Variables
Two
Factors
Two-way
ANOVA H0:
A: µ1 = µ2 = …= µq
B: µ1 = µ2 = …= µr
AB: Interaction = 0
Test
One r
Compare
Two r’s
t-test H0:
ρ=0
z-test H0:
ρ1 = ρ2
zobs 
FA 
MS A
MSW
FB 
MS B
MSW
FAB 
MS AB Interaction
MSW ithin
df A  p  1
df B  q  1
df AB  ( p  1)( q  1)
dfW  N  pq
t obs 
robs
2
1  robs
N 2
Or compare
rcritical with robs
z r1  z r2
1
1

N1  3 N 2  3
Final Exam will be posted tomorrow
on Coursework…due December 14.
(I’ll send out an email to let you know it’s there.)
Thanks for a great quarter!!