1. Inference for Regression Chapter 14

2. Linear Regression We can use least squares regression to estimate the linear relationship between two quantitative variables. Using this relationship we can predict the response variable y given the explanatory variable x in the equation y=a+bx. a and b are statistics. (How is that different from parameters?_

3. Analyzing data Plot and interpret. Look for form, direction and strength as well as outliers and other deviations. Numerical summary: If the data shows a roughly linear pattern, the correlations describes the direction and strength of the relationship] Mathematical model: Find least-squares regression line for predicting y given x.

4. Thinking about the model Slope b and intercept a are statistics. Different sample of data may have led to different values for b and a. Formal inference requires us to think of a and b as estimates of parameters  and .

5. Conditions for performing inference on regression model For any fixed value of x the response y varies according to a normal distribution. The mean response  y has a straight-line relationship with x:  y =  +  x The standard deviation of y (call it  ) is the same for all values of x. The value of  is unknown.

6. The regression model

7. Inference First step is to estimate ,  and  The slope b of the least-squares line is an unbiased estimator of the true slope  and the intercept a of the least squares line is an unbiased estimator of the true intercept . Use s to estimate the unknown  of the model.

8. Calculating s After entering data in calculator use LinRegTTest which will calculate linear regression equation and s (as well as more things that we’ll get to.)

9. Confidence Interval Run test LinRegTint. Gives you confidence interval for  Page 789 gives you an equation Degrees of freedom for regression is n-2. (Since there are now two variables we lose 2 degrees of freedom.)

10. Interpreting computer output Generally questions about inference are accompanied with computer generated output.

11. Inference Testing We will test the hypothesis about the slope . Generally this is H o :  =0 This would say that the mean of y does not change at all when x changes – or that there is no true linear relationship between x and y. Regression output generally gives p for a two- sided test. If you are doing a one-sided test, divide p by 2.

12. Backpack weight

13. Linear Regression t-test Parameter:  :the slope of the population regression relating student weight to backpack weight. Hypotheses: Ho: β=0; Ha: β≠0 Conditions: scatterplot indicates linear relationship; SRS; errors around each value of x follow a normal distribution; population at least 10 times sample

14. Linear Regression t-test From the output: t=3.21; p =.018 Since p=.018 is small (Let α=.05) we reject the null hypothesis. There is evidence that there is a linear relationship between student’s body weight and their backpack weight.

15. Now you try Is there a linear relationship between weight and length of bears?