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### Last Time:Finished Contingency TablesReviewed Basics on Linear Regression

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

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 people:

Finding a p-value for a Chi-Square Statistic (one-tailed)

Table F (T-20)

Example: Development in 1st grade

Example: Development in 1st grade

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.

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 variable)

Hypothesis Testing:

Example 20 kindergarteners

“Social Competence Score”

Statistical Inference (for two variables)

Example:

Children X: Popularity,

Y: Social competence

Goal: Explain (linear) relationship between X and Y

Statistical Inference (for two variables)

Simple (linear) regression

Explain the (linear) relationship (if it exists)

between random variable X and random variable Y.

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

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: value of X(just a change of notation)

Estimated Regression Line value of X

Recall: Point Estimates value of X(Sample Statistics) are Random Variables

Recall: Point Estimates value of X(Sample Statistics) are Random Variables

Sampling Distributions

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 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

Hypothesis Test on Slope value of X

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 Regression value of X

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 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 Table value of X

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

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 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 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… value of XLast Time…This Time…

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