Inference for Regression

1. Moderately strong, positive, linear association No apparent outliers. 7. For every increase of 1 mile in the distance there tends to be an increase of $0.117 in the price of the ticket. 2. 8. For every increase of 100 miles there tends to be an increase of $11.70 in the price of the ticket. 3. 9. 63.2% of the variability in the price of the ticket can be explained by variations in the distance. 10. 4. Underestimate 5. 6. Not a good prediction. Outside the range of the data and is an extrapolation. 12. 11. Since the residuals are scattered and no apparent curve is present the linear model is an appropriate one. 13.Given the following regression output, what is the LSRL equation? 14. What is the correlation? Slope is negative so r = -0.308 • We want to test if there is an association between two quantitative variables (x and y) • Least Squares Regression Line: Where: Inference for Regression Regression Slope t-Test • Where ε is the natural errors that occur for each point. • The errors are estimated by the residuals, e. • We use the residuals to help us determine if a linear model is appropriate. We look for a curve in the residual plot. • The residuals are independent and normally distributed; N(0, se) • Estimates the linear relationship between variables from a sample of a population. • b0 and b1 are unbiased estimators of the population parameters β0 and β1 • The idealized model then becomes: • To test for an association we check β1. • If no association exists the slope of the regression line should be 0. • It can be shown that b1 has a Student’s tdistribution with n – 2 degrees of freedom • Where • Hypothesis: H0: β1 = 0 There is no association between… HA: β1 ≠ 0 There is an association between… Conditions: • Straight Enough Condition: Check scatterplot that there is no obvious curve or large outliers • Independence: data can be assumed to be independent • Equal Variance: Check the residual plot for changes in the spread of the residuals • Nearly Normal Condition: Create a histogram or Normal Probability plot of the residuals All conditions have been met to use Student’s tmodel for a regression slope t-test. • A computer output is given that we use to run the test. y-variable b0 x-variable P-Value b1 R2 SE(b1) se Mechanics: df = n – 2 Conclusion: If reject H0, there is evidence of an association • For our work we rarely do any mechanics by hand. Example: Does a relationship exist between High School GPA and freshman year performance in college? A random sample of 40 freshman at a local college was taken and their HSGPA and the GPA from their first full year were recorded. • Hypothesis: H0: β1 = 0; No association exists between HS GPA and freshman GPA HA: β1 ≠ 0; An association exists between HS GPA and freshman GPA Conditions: 1. Independence: It is safe to assume each student is independent of the next student 2. Straight enough condition: no obvious bend or curve exists in the scatterplot. 3. Equal Variance: The residual plot shows no change in the variance 4. Nearly Normal Condition: the normal probability plot of the residuals shows an approximately linear shape All conditions have been met to use Student’s t-model for a regression slope t-test. Mechanics: df = 40 – 2 = 38; α = 0.05 Since the P-Value is less than alpha (4.67 x 10-6 <0.05) I reject the null hypothesis. There is statistically significant evidence of an association between HS GPA and freshman year GPA. Confidence Interval All conditions have been met to use Student’s tmodel for a regression slope t-interval. Formula: 95% Confidence interval for the slope of the regression line is: I am 95% confident that on average for every increase of 1 point in HSGPA there is an increase of between 0.676 to 1.504 points in first year GPA. Using TI LinReg Test for AIRFARE DATA: t-statistic P-Value df Slope Se SE(b1) is not given in the output from the calculator. To find it divide slope by t