Statistics Lecture 21

zLast Day: Introduction to Regression zToday: More Regression zAssignment: 11.38, 11.41, 11.68

Quick Review zOften concerned with describing the relationship between two quantitative variables X and Y zCan visualize relationship with scatter-plots zCan measure strength and direction of linear association by the correlation coefficient, r zWhen there is a linear relationship between a response variable and a predictor variable, can use least squares regression

Example (cont.) zData: The table below gives the average radioactivity in milk samples and the percent increase in the number of deaths for 9 regions of the U.S.A.

Residuals zWe do not know the “true” line describing the relationship between X and Y zLine was estimated, using the data, by the method of least squares zIf the line fits well, the data points should fall close to the line, in general zResiduals:

zCan use residuals to estimate the variability about the straight line zAlso called MSE.

Inference for Least Squares Estimators zThe least squares regression line computed from the data is an estimate of the “true” straight line zIf we had another set of responses, Y, at the same X values, would they be the same? Why? zWould we observe the same values? Why?

Standard Errors for the Slope and Intercept zStandard Error of the Slope: zStandard Error of the Intercept:

Inference for the Slope zA fundamental and important question is whether there really is a significant linear relationship between X and Y zIf there is no linear relationship between X and Y, what would the slope be? zIf there is a linear relationship between X and Y, what would the slope be? zIf there is a positive linear relationship between X and Y, what would the slope be? zIf there is a linear relationship between X and Y, what would the slope be?

Hypothesis for the Slope zHypotheses: zTest Statistic: zP-Value:

Confidence Interval for the Slope:

Example (radiation): zIs there a significant LINEAR relationship between radiation and the increase in deaths?

Inference for the Intercept zSomewhat less important than the slope zCan be interested in testing hypotheses for the intercept

Hypothesis for the Intercept zHypotheses: zTest Statistic: zP-Value:

Confidence Interval for the Intercept:

Example (radiation): zTest the hypothesis that the intercept is 0 with significance level 0.05 z95% Confidence Interval

Estimation of the Mean Response at a Specified x zWhy would we want to compute the regression line? zEstimated mean response at a specified x zPrediction of a future response at a specified x z What is the difference?

CI for the Mean Response at a Specified x zA confidence interval for the mean response is:

Prediction Interval for a Single Response at a Specified x zA prediction interval for a future response is:

Example (radiation): zGive a 95% confidence interval for the mean response when the radiation is 24 picocuries/L zGive a 95% prediction interval for the a single response when the radiation is 24 picocuries/L