STP 311 - Module 2 Learning Lab

1. Recall from Learning Lab #1, write the regression equation that allows you to predict a science score based on math performance. y = science x = math ^ y = ^ β 0 + ^ β 1 x = 16.8 + 0.67 x 2. Use SAS (option clb) to find a 95% confidence interval for β1. Interpret your interval. Parameter Estimates Variable DF Parameter Estimate Standard Error t Value Pr > |t| 95% Confidence Limits Intercept 1 16.75789 3.11623 5.38 <.0001 10.61264 22.90315 math 1 0.66658 0.05828 11.44 <.0001 0.55165 0.78151 The 95% confidence interval for 𝛽 1 is (0.55, 0.78). Thus, we are 95% confident that the true slope is between 0.55 and 0.78. That is the true change in science score for every 1 unit increase in math score is between 0.55 and 0.78. 3. Suppose we want to test the hypothesis H 0 : 𝛽 1 = 0 vs H 1 : 𝛽 1 ≠ 0 a. Without using statistical jargon, explain what you are testing in the context of the problem. We are trying to test whether there is a relationship between science score and math score. b. What is the p-value? Parameter Estimates Variable DF Parameter Estimate Standard Error t Value Pr > |t| 95% Confidence Limits Intercept 1 16.75789 3.11623 5.38 <.0001 10.61264 22.90315 math 1 0.66658 0.05828 11.44 <.0001 0.55165 0.78151 c. What can you conclude about the relationship between science and math scores? Is it reasonable to use math scores to predict science scores? H 0 : 𝛽 1 = 0 : There is no linear relationship between science scores and math scores H 1 : 𝛽 1 ≠ 0 : There is a significant linear relationship between science score and math score. Since the p-value is < 0.001, we can conclude that β1 ≠ 0 and there is a significant linear relationship between science scores and math scores. Hence, we can predict science scores based on math scores.

4. Use the ANOVA table to perform an F test for the significance of the straight line relationship. Interpret result. Analysis of Variance Source DF Sum of Squares Mean Square F Value Pr > F Model 1 7760.55791 7760.55791 130.81 <.0001 Error 198 11747 59.32799 Corrected Total 199 19507 H 0 : 𝛽 1 = 0 : There is no linear relationship between science scores and math scores H 1 : 𝛽 1 ≠ 0 : There is a significant linear relationship between science score and math score. The F-value is 130.81 and the p value of the test is <0.0001 meaning that there is a significant linear relationship between population density and average insurance rates. 5. Verify F = t 2 using the values from the output. The F value is 130.81 and the t 2 is 11.44 2 = 130.87 6. Your friend makes a claim that for every point you increase your math score, your average science score increases, on average, 0.7 points! Use your answer in #2 as well as the relationship between a hypothesis test and confidence interval to substantiate or refute your friend's claim. H 0 : 𝛽 1 = 0.7 H 1 : 𝛽 1 ≠ 0 .7 And since the confidence interval (0.55, 0.78), 0.7 falls within the interval, that means theat we fail to reject the null hypothesis that 𝛽 1 = 0.7 at 95% confidence level. There is no sufficient evidence that there for every point increase in math scores, the science score increases by 0.7 points. Notes: Code