Section 11.5 Inferences Concerning the Regression Coefficients

Example 11.2

Confidence Interval for b , the slope parameter

A (1-a)100% confidence interval for the parameter b in the regression line mY|x  =  a + bx  is

where  is a value of the t-distribution with n-2 degrees of freedom

## Example 11.2

Hypothesis testing on the slope parameter, b

To test the null hypothesis Ho:b = bo against any suitable alternatives,

use t-distribution with n-2 degrees of freedom to define the critical region, and the following test statistic

## Minitab Example:

Statistical Inference on the intercept

Confidence Interval for a , the intercept parameter

A (1-a)100% confidence interval for the parameter a  in the regression line mY|x  =  a + bx  is

or

where  is a value of the t-distribution with n-2 degrees of freedom

## Example 11.4

Hypothesis testing on the intercept parameter, a

To test the null hypothesis Ho: a = ao against any suitable alternatives,

use t-distribution with n-2 degrees of freedom to define the critical region, and the following test statistic

## Example 11.5

df=33-2=31, P-value =

From Table A.4 directly, P-value < 0.05. So, the null hypothesis of zero intercept is rejected at 0.05 level of significance.

A measure of quality of fit: Coefficient of Determination, R2

Coefficient of Determination, R2 = proportion of total variability in the dependent variable Y explained by the fitted model.

TSS = SSR + SSE

Coefficient of Determination,

R2 = 1.0                 if fit is perfect

R2 = 0.0                 if fit is poor

Section 11.6 Prediction

Confidence Interval for  , the mean response

A (1-a)100% confidence interval for the mean response   in the regression line mY|x  =  a + bx  is

where  is a value of the t-distribution with n-2 degrees of freedom

## Example 11.6

Prediction Interval for  , the future value of a response

A (1-a)100% prediction interval for a single response  in the regression line mY|x  =  a + bx  is

where  is a value of the t-distribution with n-2 degrees of freedom

## Example 11.7

Section 11.11  Simple Linear Regression Case Study

A more complicated model may be more appropriate

The higher R2 value would suggest that the transformed model is more appropriate.

Section 11.12 Correlation

Correlation coefficient

The measure r of linear association between two variables X and Y is estimated by the sample correlation coefficient r, where

-1 <  r < 1

r = 1.0 or –1.0       perfect linear relationship

r = 0.0                   no linear relationship

Sample coefficient of determination

represents the proportion of the variation in TSS explained by the regression of Y on x, namely, SSR.

## Example 11.10

Hypothesis testing on the correlation coefficient, r

To test the null hypothesis Ho: r = ro against any suitable alternatives,

We can use t-distribution with n-2 degrees of freedom to define the critical region, and the following test statistic

but this test has problem when r values close to -1 or 1

However, a more general test is given by

the approximate normal distribution with mean and variance .

We can use the standard normal to define the critical region, and the following test statistic

## Example 11.12

Important: Correlation is a measure of linear relationship, so r = 0 does not necessarily mean there is no relationship between two variables.