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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


Last time finished contingency tables reviewed basics on linear regression

Last Time:Finished Contingency TablesReviewed Basics on Linear Regression


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!


Suppose i sample n many people2
Suppose I sample n many people:






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.



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


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:

±

·


Statistical inference for a single variable1
Statistical Inference (for a single variable)

Hypothesis Testing:


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


Statistical inference for two variables1
Statistical Inference (for two variables)


Y

1

X


Simple linear regression
Simple (linear) regression

Explain the (linear) relationship (if it exists)

between random variable X and random variable Y.


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


First assumption error has a normal distribution
First Assumption:Error has a Normal Distribution

Error




Y value of X

X


Y value of X

X


We will sample data to estimate the parameters. value of X

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.


Estimation of intercept and slope just a change of notation
Estimation of Intercept and Slope: value of X(just a change of notation)



Parameter estimates
Parameter Estimates: value of X

Degrees of freedom

loose 1 df for X

loose 1 df for Y


Recall point estimates sample statistics are random variables
Recall: Point Estimates value of X(Sample Statistics) are Random Variables


Recall point estimates sample statistics are random variables1
Recall: Point Estimates value of X(Sample Statistics) are Random Variables

Sampling Distributions


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

Don’t

Know!

Hypothesis Testing


Remember the general rule for Confidence Intervals: value of X

Point

estimate

critical

value

Std. dev. of

point estimate

±

·


Confidence intervals for intercept and slope
Confidence Intervals for value of XIntercept and Slope

Point

estimate

critical

value

Std. dev. of

point estimate

±

·




Hypothesis Test on Slope value of X

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

and conclude that there is indeed a linear relationship



Computer output note different programs differ in style and content
Computer Output value of X(Note: Different programs differ in style and content!)

p-value

< .001


Analysis of variance for regression
Analysis of Variance value of Xfor 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 value of X

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 value of X

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)


Analysis of variance for regression4
Analysis of Variance for Regression value of X

Mean Squares

Error

(MSE)



Hypothesis testing and the anova table
Hypothesis Testing value of Xand 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 value of X

Table E

df in the numerator

df in the denominator


Analysis of variance table2
Analysis of Variance Table value of X

p

p-value here: <.001


Note value of X


The Square of a value of Xt 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 value of X

  • 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 value of Xbetween 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.


loose 1 df for value of XX

loose 1 df for Y

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

and conclude that there is indeed a linear relationship


In our Example: value of X


Hmmmm last time this time
Hmmmm… value of XLast Time…This Time…


One more thing
One more thing… value of X

Percent of Variance

explained by the model


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