WARM – UP Can you accurately predict your TEST grade based on your QUIZ grade? STUDENT QUIZ TEST 1.Describe the association between Quiz Grades and Test Grades. 2.Write the equation of the line of regression. 3.Use this model to predict a Test Grade from a Quiz grade of Is this a good model?

QUIZ Grade TEST Grade QUIZ Grade Residuals TEST = (QUIZ) = (50) = 1.Describe the association between Quiz Grades and Test Grades 2.Write the equation of the line of regression. 3.Use this model to predict a Test Grade from a Quiz grade of Is this a good model? STUDENT QUIZ TEST

Regression Inference C.I. To estimate the rate of change/slope, we use a Confidence Interval. The formula for a confidence interval for  1 is:

WARM – UP STUDENT QUIZ TEST 1.Find the 95 % Conf. Interval for the Rate of Improvement from the Quiz to the Test. We can be 95% confident that for every additional point increase in quiz grade your test grade is predicted to be to points higher. Dependent Variable is: Test R-squared = 35.3% s = with 8 – 2 = 6 degrees of freedom Variable Coefficient SE(Coeff) T-ratio P-Value Intercept Quiz

LINEAR REGRESSION t – TEST

Is there a relationship between the amount of lead verses the amount of Zinc present in fish? State Hypothesis and Conclusion. Lead Zinc Because the p-val < 0.05 Reject Ho. There is evidence to conclude that a relationship exists b/t lead and zinc content in fish. Dependent Variable is: Zinc R-squared = 85.7% s = with 8 – 2 = 6 degrees of freedom Variable Coefficient SE(Coeff) T-ratio P-Value Intercept Lead

P-value Is there a relationship between the amount of lead verses the amount of Zinc present in fish? State Hypothesis and Conclusion. Metal Concentrations in 1999 Spokane River Fish Samples (concentrations in parts per million (ppm) Source: WA State Dept. of Ecology report) Lead Zinc Because the p-val < 0.05 there is significant evidence to conclude that a relationship exists b/t lead and zinc content in fish. Se b = =

Estimate the Rate of change for the relationship between lead and zinc using a 99% Confidence Interval Metal Concentrations in 1999 Spokane River Fish Samples (concentrations in parts per million (ppm) Source: WA State Dept. of Ecology report) Lead Zinc We can be 99% confident that the true slope of the relationship between lead and zinc is between. Dependent Variable is: Zinc R-squared = 85.7% s = with 8 – 2 = 6 degrees of freedom Variable Coefficient SE(Coeff) T-ratio P-Value Intercept Lead

P-value Is there a relationship between the amount of lead verses the amount of Zinc present in fish? State Hypothesis and Conclusion. Metal Concentrations in 1999 Spokane River Fish Samples (concentrations in parts per million (ppm) Source: WA State Dept. of Ecology report) Lead Zinc Because the p-val < 0.05 there is significant evidence to conclude that a relationship exists b/t lead and zinc content in fish. Dependent Variable is: Zinc R-squared = 85.7% s = with 8 – 2 = 6 degrees of freedom Variable Coefficient SE(Coeff) T-ratio P-Value Intercept Lead

Is there a relationship in the amount of lead verses the amount of Zinc present in fish? Performing a Linear Regression t-Test from the Calculator Lead Zinc Because the p-val < 0.05 there is significant evidence to conclude that a relationship exists b/t lead and zinc content in fish. STAT  Test  LinRegTTest

EXAMPLE: Manatees living off the coast of Florida are often killed by powerboats. Here are data from 1984 – 1990 Powerboat Registrations In Thousands Manatees Killed Find the 95% confidence interval for the rate at which the Manatee deaths change with regards to powerboat registration. Dependent Variable is: Manatees R-squared = 98.2% s = with 7 – 2 = 5 degrees of freedom Variable Coefficient SE(Coeff) T-ratio P-Value Intercept Boats < We can be 95% confident that for every additional 1000 powerboat registered, between and more Manatees are killed.

Show that the p-value is by using the t formula and calculating the p-value. Lead Zinc Dependent Variable is: Zinc R-squared = 85.7% s = with 8 – 2 = 6 degrees of freedom Variable Coefficient SE(Coeff) T-ratio P-Value Intercept Lead SE(b 1 )