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Homework Assignment. 10.17, 10.19, 10.23, 10.29, 10.32 Due in Class Dec 1. Last Time: Finished Contingency Tables Reviewed Basics on Linear Regression. Suppose I sample n many people:. How many observations do I expect to get in cell (i,j)?

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homework assignment

Homework Assignment

10.17, 10.19, 10.23, 10.29, 10.32

Due in Class Dec 1

suppose i sample n many people
Suppose I sample n many people:
  • How many observations do I expect to get in cell (i,j)?
  • If the Null Hypothesis holds, i.e., if the columns and rows are independent, then I expect the number of observations in cell (i,j) to be

How to compare??

Both are unknown!

suppose i sample n many people1
Suppose I sample n many people:

How to compare??

Both are unknown!

slight change of notation
Slight Change of Notation

Homogeneity of

parallel samples

example
Example

255

32

FIXED

Equivalent Equations

more generally
More Generally:

The rest is the same as in previous scenario,

i.e., we get the same Chi-square again.

slide17
The Square of a Standard Normal Random Variableis a Chi-Square Random Variablewith 1 degree of freedom.
slide18

Today:From descriptive to inference statistics…Estimation and Hypothesis Testingfor Linear Regression

statistical inference for a single variable
Statistical Inference (for a single variable)

Estimation: (Confidence Intervals)

Point

estimate

critical

value

Std. dev. of

point estimate

±

·

For instance: Confidence Interval for the mean:

±

·

example 20 kindergarteners
Example 20 kindergarteners

1pt

2pts

3pts

“Popularity Score” = Average Score

example 20 kindergarteners1
Example 20 kindergarteners

“Social Competence Score”

example 20 kindergarteners2
Example 20 kindergarteners

“Popularity Score”

“Social Competence Score”

statistical inference for two variables
Statistical Inference (for two variables)

Example:

Children X: Popularity,

Y: Social competence

Goal: Explain (linear) relationship between X and Y

slide26
Y

1

X

simple linear regression
Simple (linear) regression

Explain the (linear) relationship (if it exists)

between random variable X and random variable Y.

slide28
Four assumptions

about the error term

4

For different

values of X,

the error terms

are uncorrelated

1

3

2

The error term

is a normally

distributed

random variable

No matter what

value X takes,

the error has

a mean of zero

slide32
Y

X

slide33
Y

X

slide34
We will sample data to estimate the parameters.

This leads to point estimates, confidence intervals

and hypothesis testing for each parameter,

in addition to a general test of the model as a whole.

parameter estimates
Parameter Estimates:

Degrees of freedom

loose 1 df for X

loose 1 df for Y

recall point estimates sample statistics are random variables2
Recall: Point Estimates (Sample Statistics) are Random Variables

Don’t

Know!

Hypothesis Testing

slide43
Remember the general rule for Confidence Intervals:

Point

estimate

critical

value

Std. dev. of

point estimate

±

·

confidence intervals for intercept and slope
Confidence Intervals for Intercept and Slope

Point

estimate

critical

value

Std. dev. of

point estimate

±

·

slide47
Hypothesis Test on Slope

If p-value of the standardized statistic <  then reject H0

and conclude that there is indeed a linear relationship

analysis of variance for regression
Analysis of Variancefor Regression

How much

y differs

from mean

How much

predicted y

differs

from mean

residual

/

error

Involves

only data

analysis of variance for regression2
Analysis of Variance for Regression

Sum Squares

Total

(SST)

Sum Squares

Error

(SSE)

Sum Squares

Model

(SSM)

Variability

in the

Data

Variability

unaccounted

for

Variability

accounted for

by the Model

Degrees of Freedom

DFT = n-1

Degrees of Freedom

DFE=n-2

Degrees of Freedom

DFM = 1

=

+

analysis of variance for regression3
Analysis of Variance for Regression

Degrees of Freedom

DFT = n-1

Degrees of Freedom

DFE=n-2

Degrees of Freedom

DFM = 1

Mean Squares

Total

(MST)

Mean Squares

Error

(MSE)

Mean Squares

Model

(MSM)

hypothesis testing and the anova table
Hypothesis Testingand the ANOVA Table

Mean Squares

Error

(MSE)

It can be shown that

the Null Hypothesis

implies that

MSM is also an unbiased

estimator of

analysis of variance table1
Analysis of Variance Table

Table E

df in the numerator

df in the denominator

analysis of variance table2
Analysis of Variance Table

p

p-value here: <.001

slide60
The Square of a t Random Variable with n-2 degrees of freedomis an F Random Variablewith 1 degree of freedom in the numerator andwith n-2 degrees of freedom in thedenominator.
slight shift of perspective
Slight shift of perspective
  • Treated the X variable as if it were fixed.
  • Now, let’s think of X and Y as both being random variables (jointly distributed).
  • Let’s assume they are both normally distributed (i.e. they are “jointly normal”)
  • We can define a population correlation ρ
  • We can use the sample correlation r as an estimate of ρ.
is there a linear relationship between random variables x and y
Is there a linear relationship between random variables X and Y?

Hypothesis Test about Correlation

Question: Can we / can we not draw a line close to the data?

Answer: No, unless we provide sufficient evidence that we can.

slide63
loose 1 df for X

loose 1 df for Y

If p-value of the standardized statistic <  then reject H0

and conclude that there is indeed a linear relationship

one more thing
One more thing…

Percent of Variance

explained by the model

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