Linear Regression: Inference and Hypothesis Testing, Exams of Mathematics

Download Linear Regression: Inference and Hypothesis Testing and more Exams Mathematics in PDF only on Docsity! Statistical Applications ACTIVITY 8: Inference for Linear regression Why For any set of points there is a line of best fit, with a slope and an intercept (just as any set of numbers has a mean and a standard deviation). for most really data, though, the point of collecting the data is to draw conclusions about a population from which the data were drawn. We want to extend the tools of inference (estimation — with confidence — and testing for significance) to the relation between variables - particularly the slope, which gives (ideally) the rate of change of (the mean of) the response as the predictor changes. LEARNING OBJECTIVES 1. Work as a team, using the team roles 2. Gain experience with linear regression concepts 3. Be able to use and interpret tests of significance for linear regression. 4. Be able to find and intepret confidence interval estimates for the slope of a regression line. CRITERIA 1. Success in completing the exercises. 2. Success in answering questions about the model 3. Success in working as a team RESOURCES 1. The “Simple Linear Regression” handout from last week - especially the sections on testing the regression coefficient, on confidence intervals for β1, on the F-test. 2. Your Text - especially sections 14.4 - 14.5 and the statistical tables 3. Your Calculator 4. 40 minutes PLAN 1. Select roles, if you have not already done so, and decide how you will carry out steps 2 and 3 (5 minutes) 2. Work through the exercises given here - be sure everyone understands all results (30 minutes) 3. Assess the team’s work and roles performances and prepare the Reflector’s and Recorder’s reports including team grade (5 minutes). 4. Be prepared to discuss your results. EXERCISE 1. A sales manager collected the following data on annual sales and years of experience for 10 salespeople: Years of Annual Sales Salesperson Experience ($1000s) 1 1 80 2 3 97 3 4 92 4 4 102 4 6 103 6 8 111 7 10 119 8 10 123 9 11 117 10 13 136 1 (a) Plot these data with Years of experience as the independent variable. Does there appear to be a linear relationship? (allowing for random variation, of course) (b) Give (use your calculator) the regression line for predicting annual sales based on years of experience and plot the line on your graph. [To check your data entry: x̄ = 7, ȳ = 108] (c) On the average, how much does the annual sales amount increase for each year of experience? (d) For this group of people, how much of the variation in annual sales is explained by differences in experience? (e) What does your equation predict as the average annual sales for a person with 10 years experi- ence? What is the residual (the error of prediction) for person #7 (who has 10 years experience)? 2. At Huge University, a study was done to establish whether a relationship existed between a a student’s SAT score (at the time of application) and GPA (at the time of graduation). Here is part of the Minitab printout for a regression based on ten students. Regression Analysis: GPA versus SAT The regression equation is GPA = 0.977 + 0.00337 SAT Predictor Coef SE Coef T P Constant 0.9772 0.9123 SAT 0.003368 0.001492 S = 0.374380 R-Sq = 38.9% R-Sq(adj) = 31.3% Analysis of Variance Source DF SS MS F P Regression 1 0.7147 0.7147 Residual Error 8 1.1213 0.1402 Total 9 1.8360 (a) Set up and carry out the test to determine whether this gives evidence of a linear relation between entering SAT and graduating GPA (at .05 level). (Note: there is enough information here to use the F–test, but if you want to use t, you need to know that the sum of squares of the SAT scores is 63002.) (b) Use the information to give a 95% confidence estimate for the increase in (average) GPA for each one point increase in SAT score. (c) The data set that produced the results given below gives the same regression line, and same R2 value as the first set. Carry out the test to see if this data set would give evidence (at the 5% level) of a linear relation between SAT and GPA. (For this set, sum of squares of SAT scores is 189007). As you can see, r2 (or R2, as presented by Minitab) does not determine significance. The regression equation is C6 = 0.977 + 0.00337 C5 Predictor Coef SE Coef T P Constant 0.9772 0.4877 C5 0.0033681 0.0007973 S = 0.346609 R-Sq = 38.9% R-Sq(adj) = 36.7% Analysis of Variance 2