EDUC 200C

EDUC 200C

week10December 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 H0Type I error

αCorrect!

Power: 1-β

Retain H0Correct!

Confidence: 1-αType 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

Mean

Variability

Standard Deviation

Variance

Correlation Regression

Central Tendency

One Variable

Two Variables

XbYa X

Y

ss

rb

1

N

ZZr

YX

1

)( 22

N

XXs

1

)( 2

N

XXs

Slope Intercept Pearson

Correlation Coefficient

(interval/ratio) N

XX

Descriptive Statistics

Relative Standing

Z score Percentile Rank

or

sXX

z

Xz

Frequency

Mode Median

)1(

61 2

2

NN

Drs

Spearman Correlation Coefficient (Ordinal)

The value of the middle case (if N is odd)

The average of the values of the two

middle cases (if N is even)

The most frequent Xi

Concept Map: Descriptive

t-test H0: µ = Constant

One-way ANOVA H0:

µ1 = µ2 = µp = µ

Two Independent

Groups

Two Paired Groups One

Factor Two

Factors

More than Two Group

One Group

Two Groups

Compare Means

Two-way ANOVA H0:

A: µ1 = µ2 = …= µq B: µ1 = µ2 = …= µr AB: Interaction = 0

Population SD Known

Population SD Unknown

t-test H0: µ1 = µ2

t-test H0: µD = 0

z-test H0: µ = Constant

1/

t

NdfNs

D

D

gNdfgdfMSMSF

W

B

Within

Between

1 1/

NdfNs

Xt

2

11*

2)1()1(

21

2121

222

211

21

NNdf

NNNNsNsN

XXt

NXz/

pqNdfqpdf

qdfpdfMS

MSF

MSMSF

MSMSF

W

AB

B

A

Within

nInteractioABAB

W

BB

W

AA

)1)(1(11

Examine Associations

Correlation between Two

Variables

Test One r

Compare Two r’s

31

31

21

21

NN

zzz rrobs

Or compare rcritical with robs

21 2

Nr

rt

obs

obsobs

Inferential Statistics

z-test H0: ρ1 = ρ2

t-test H0: ρ = 0

Concept Map: Inferential

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!!