Lecture 37

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The coefficient of determination (r-squared) is a indicator of whether or not there is a good fit (or relationship) between the two variables

r2 = SSR/SSTO = 1 - SSE/SSTO

Value of r-squared always ranges from 0 to 1

What is the value of r-squared if there is a perfect fit between two variables?

What is the value of r-squared if there is no relationship between two variables?

Coefficient of correlation {r} is simple the square root of the coefficient of determination (r-squared)

Some prefer to present their results in terms of r rather that r-squared

This can be a little deceptive since r < r-squared

The value of r ranges from -1 to +1

The sign of r is the same as that of b1

 

 

 

Hypothesis testing is conducted to determine where or not the slope of the regression line is zero

The hypothesis that is tested is

 

Ho: beta1 = 0

H1: bets1 not equal 0

 

If we conclude Ho, does this mean that we have a strong relationship between X and Y?

 

This hypothesis test is really a test for whether or not there is a statistically significant relationship between X and Y

The F-test

Two different procedures can be used to test the above hypothesis - the F-test and the t-test

Both procedures are related

F-test

The F-test is based MSR/MSE

If MSR/MSE is 1, then b1 = 0

The reason is that

 

If F*=MSR/MSE is large, reject the null hypothesis

What is large?

The sampling distribution of F* is based on the F-distribution

The relevant F-statistic is F(1-a;1, n-2)

(note F has two degrees of freedom - df for the numerator and df for the denominator)

If F*>F(1-a;1, n-2) reject the null hypothesis, accept H1 (there is a significant relationship between X and Y)

 

The t-test

Test the same hypothesis as F-test

The t-test is based on the sampling distribution of b1

Mean

Variation

Distribution Type: Normal Distribution

 

Estimated variance of b1 is

 

 

 

The test statistic is

t* = (b1 - 0)/s{b1}

If t* > t(1-a/2; n-2) reject null hypothesis

Lecture 38

Confidence Interval for Mean Response

One use of regression analysis is to predict the mean value of Y for a given value of X

For example, what is the average cost when the sales level is 800

The predict average is written as Y-hat

The specified level of X is X_h

 

Is Y-hat a random variable? Why?

 

What are the parameters needed to characterize the sampling distribution?

Mean

Variation

 

 

 

Distribution Type

Normal Distribution

In practice, B0, B1 and s2 are not know

The estimators for B0 and B1 are b0 and b1

The for s2 is MSE

Therefore, the estimator for s2 {Y-hat h) is s2 {Y-hat h)

 

 

Shortcut for above equation

Based on the sampling distribution the (1-a) confidence interval of the mean response is

 

 

 

Where t = t(1-a/2: n-2)

 

 

 

Factors Affecting Variance of the Mean Response

The variance of the mean response (and hence the width of the confidence interval) varies with the following parameters

I. The MSE - the larger MSE, the larger the variance of the mean response

 

II. Deviation of Xh from X-bar - the further Xh is from X-bar the larger the variance of the mean response.

From a practical point of view this means that, in planning a study, we should try to make sure that the mean value of the observed Xis is close to the value of X which is of most significance in the study

 

III. Variability of Xi’s - the greater the variability of Xis, the smaller the variance of the mean response. In planning our study, we want as wide a range as possible for the Xi’s

 

IV. Size of Sample - the larger the sample size the smaller is the variability of the mean response

 

Lecture 39