1. WARM – UP Can you accurately predict your TEST grade based on your QUIZ grade? 12345678 88841009270808192 951001029490859997 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 50. 4.Is this a good model?

2. QUIZ Grade TEST Grade QUIZ Grade Residuals TEST = 64.217 + 0.361 (QUIZ) = 64.217 + 0.361(50) 82.286 = 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 50. 4.Is this a good model? 12345678 88841009270808192 951001029490859997 STUDENT QUIZ TEST

3. 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:

4. WARM – UP 12345678 88841009270808192 951001029490859997 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 - 0.1276 to 0.8504 points higher. Dependent Variable is: Test R-squared = 35.3% s = 4.866 with 8 – 2 = 6 degrees of freedom Variable Coefficient SE(Coeff) T-ratio P-Value Intercept 64.2170 14.204 1.6249 0.1386 Quiz 0.36137 0.19985 1.8082 0.1206

5. LINEAR REGRESSION t – TEST

6. Is there a relationship between the amount of lead verses the amount of Zinc present in fish? State Hypothesis and Conclusion. Lead Zinc 0.73 45.3 1.14 50.8 0.60 40.2 1.59 64.0 4.34 150. 1.98 106. 3.12 90.8 1.80 58.8 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 = 15.3330 with 8 – 2 = 6 degrees of freedom Variable Coefficient SE(Coeff) T-ratio P-Value Intercept 23.0796 14.204 Lead 27.5335 4.5837

8. 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 0.73 45.3 1.14 50.8 0.60 40.2 1.59 64.0 4.34 150. 1.98 106. 3.12 90.8 1.80 58.8 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 = 4.5837 = 0.0010

9. 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 0.73 45.3 1.14 50.8 0.60 40.2 1.59 64.0 4.34 150. 1.98 106. 3.12 90.8 1.80 58.8 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 = 15.3330 with 8 – 2 = 6 degrees of freedom Variable Coefficient SE(Coeff) T-ratio P-Value Intercept 23.0796 14.204 1.6249 0.1386 Lead 27.5335 4.5837 6.0068 0.0010

10. 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 0.73 45.3 1.14 50.8 0.60 40.2 1.59 64.0 4.34 150. 1.98 106. 3.12 90.8 1.80 58.8 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 = 15.3330 with 8 – 2 = 6 degrees of freedom Variable Coefficient SE(Coeff) T-ratio P-Value Intercept 23.0796 14.204 1.6249 0.1386 Lead 27.5335 4.5837 6.0068 0.0010

11. 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 0.73 45.3 1.14 50.8 0.60 40.2 1.59 64.0 4.34 150.0 1.98 106.0 3.12 90.8 1.80 58.8 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

12. 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 45924 58533 61433 64539 67543 71152 71951 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 = 1.9391 with 7 – 2 = 5 degrees of freedom Variable Coefficient SE(Coeff) T-ratio P-Value Intercept -29.0001 14.204 1.6249 0.1386 Boats 0.1084 0.0155 6.9932 <0.0001 We can be 95% confident that for every additional 1000 powerboat registered, between 0.0686 and 0.1483more Manatees are killed.

13. Show that the p-value is 0.0010 by using the t formula and calculating the p-value. Lead Zinc 0.73 45.3 1.14 50.8 0.60 40.2 1.59 64.0 4.34 150. 1.98 106. 3.12 90.8 1.80 58.8 Dependent Variable is: Zinc R-squared = 85.7% s = 15.3330 with 8 – 2 = 6 degrees of freedom Variable Coefficient SE(Coeff) T-ratio P-Value Intercept 23.0796 14.204 1.6249 0.1386 Lead 27.5335 4.5837 6.0068 0.0010 SE(b 1 )