MATH 533 Week 6 Discussion: Regression Analysis – Download To Score An A+

MATH 533 Week 6 Discussion: Regression Analysis – Download To Score An A+ Read the Case: Statistics In Action: Legal Advertising—Does It Pay?, and answer the following questions. (The case is included in your textbook, Chapter 10.) The data set for the case study is LEGALADV, and it is available in your textbook resources, so you don't have to enter the data! Summarize what the case is about, and what the variables represent. In this week's Statistics in Action case, "Legal Advertising - Does It Pay?" is concerned with the lawsuit of two partners in a law firm over which partner should pay more of the advertising costs for the firm. Partner A, who deals with personal injury law, sued Partner B, who deals with workers' compensation law because, while the advertising is targeted more towards personal injury clients, Partner A believes that this advertising is bringing in more clients needing help with workers' compensation. The question is: is the advertising really resulting in more clients for Partner B, and should Partner B pay more for advertising? The data was collected over a 42-month period, and includes the total amount spent on advertising, including a cumulative total. The data also includes the number of new personal injury and workers' compensation clients each month. Response 2 The case presented in this week's reading is an interesting case about two former law partners who are arguing over who should pay for advertising expenses during the time they worked together. One partner, who handles personal injury cases, claims that the other partner received benefit from the advertising, despite the fact that the advertising was only for personal injury law and the other partner handled worker's compensation cases. The data presented shows the advertising expenditure per month, the number of new personal injury cases during that month, and the number of worker's compensation cases during that month. It also shows the cumulative advertising expenditures over 6 months, as presumably it may take a few months for advertising/name recognition to lead to new cases. What I find interesting is that the case asks us to look at the data for any statistically significant associations, but also asks whether we can infer that such associations actually mean causation. Response 3 This case is about legal adverting costs for a law firm and disagreements between the partners regarding who is responsible for the cost of advertising. According to this case the law firm advertising spending was focused on personal injury. The partner who only handles workers' comp claims that the partner who only handles the PI cases should play the advertising cost. The partner who handles PI cases claims that the other partner who handles WC cases also benefited from the adverting and should be responsible for the bill also. The data set we are working with represents the workers' comp and personal injury cases for 48 months, 6 months with no advertising spending and 42 months with advertising spending. Based on the given information we will need to determine which partner benefited more from advertising. Response 4 After reading this case study, I discovered that two law firm partners both handle different types of cases, one personal injury and one workers comp. Advertising is only done for personal injury cases for the firm. Partner A believes that the advertising increases cases for workers comp as well and the expenses should be shared. To me, I'm not sure this opinion makes sense as I don't know quite how these are correlated. I'm curious if you have a personal injury if it could be related to a work comp claim and still be a separate case. Or, I'm also wondering if people just see the commercial and contact the firm regarding other types of cases as well. Either way, I'd like to analyze the data and see how the two variables are related. Unfortunately I haven't yet had enough time to dig deep enough into the material to do this. Stay tuned for my next post after I take a look at the data. 1. Read pp. 559-561 and pp. 570-571 Statistics in Action. 2. Load the data from the LEGALADV file into Minitab. If you are having trouble loading the data set, see my Announcement on Data Sets for Minitab. 3. Using the Minitab Tutorials, as needed: run the scatter plots and simple regressions separately for New PI Cases on Adv. Exp. and New WC Cases on Adv. Exp. You should replicate the output on p. 571. 4. What do you see in the respective scatter plots? Are either the New PI Cases or the New WC Cases related to the Advertising Expenditures? 5. Copy and Paste your Minitab output into a Word document and include it as an Attachment. 6. For each of the simple regressions, write out the regression equation, the value of s, the standard error, and r squared, and provide their interpretations. Feel free to draw on the Week 6 Lecture to help with your interpretations. In looking at the scatterplots, it seems that there is more of an increase in the new personal injury cases, as the advertising expenditures increase, as compared with the new workers' compensation cases. The slope of regression line, when comparing new personal injury cases and advertising expenditures, is positive, indicating that the advertising has a positive effect on the number of personal injury cases. The regression line for workers' compensation and advertising expenditures is relatively flat, indicating that there is no relationship between an increase in advertising expenditures and new workers' compensation cases. General Regression Analysis: New PI Cases versus AdvExp (thousands) Regression Equation New PI Cases = 7.76751 + 0.112891 AdvExp (thousands) Coefficients Term Coef SE Coef T P Constant 7.76751 3.38499 2.29469 0.027 AdvExp (thousands) 0.11289 0.02793 4.04219 0.000 General Regression Analysis: New WC Cases versus AdvExp (thousands) Regression Equation New WC Cases = 24.5741 + 0. AdvExp (thousands) Coefficients Term Coef SE Coef T P Constant 24.5741 3.36671 7.29915 0.000 AdvExp (thousands) 0.0098 0.02778 0.35370 0.725 1. Remember, Simple Regression is essentially working with two variables at the same time and in relation to each other, but using the tools of Description and Inference that we have been using with a single variable for the past several weeks. There are an awful lot of formulas in Regression, which can be very intimidating. It is essential that we stay grounded in the core concepts of Description and Inference, measures of central tendency and dispersion, and pictures and numbers. 2. In my last post, I suggested we begin by looking at the Descriptive Statistical side of the two variable case, i.e., the Scatter Plot, the Regression Line, the Regression Equation, the Standard Error, and the Coefficient of Determination, with Interpretation. 3. Now, let's move beyond Description to Inference: See Section 10.4 in your text book, and, in particular, pp. 585-586. To get the 95% confidence intervals for the regression coefficients, use the following path: Stat: Regression: General Regression: Response: New PI Cases, Model: Adv Exp (thousands): Results: [check] Display Confidence Intervals, [uncheck] Fits and Diagnostics Table. Compare your output with that in the text book on page 586. Also, run the regression for New WC Cases and compare the output with what is in the text book, on page 586. NOTE: FOR POSSIBLE USE IN THE HOMEWORK: Under Options you can change the confidence level (from 95% to something else) and the type of confidence interval to a one tailed interval. Finally, interpret the 95% confidence intervals for the regression coefficients, for both New PI Cases and New WC Cases, using the units of the variables. 4. In case you haven't yet run the Descriptive Statistics, here is the path: Stat: Regression: fitted Line Plot: Response (Y): New PI Cases: Predictor (X): AdvExp (thousands): OK. Once you get the Descriptive regression output, compare it with what is on the fuller regression output in #3 above. 5. In sum, what's going on here is that we want to get Minitab working for us, to take care of the calculations, so we can focus on Interpretation. 6. All the best. The 95% confidence intervals for the regression coefficients for New PI Cases and New WC Cases are as follows: New PI Cases: (0.056446, 0.1693) New WC Cases: (­0.0463, 0.0660) We can be 95% confident that for every $1,000 increase in advertising expenses, the number of new personal injury cases will be between 0.056 and 0.169 cases. For the workers' compensation cases, we can be 95% confident that for every $1,000 increase in advertising expenses, the number of new worker's compensation cases will be between ­0.046 and 0.066 cases. The slope of the regression line for the workers' compensation cases is so small, showing there really is no relationship. The lower limit for the workers' compensation confidence interval is ­0.046. Could this suggest that the advertising is actually detracting from Partner B's practice? The negative in the lower limit of the confidence interval, to me, implies that Partner B's workers' compensation practice might be bringing in less cases. Results for: LEGALADV.MTW General Regression Analysis: New PI Cases versus AdvExp (thousands) Regression Equation New PI Cases = 7.76751 + 0.112891 AdvExp (thousands) 42 cases used, 6 cases contain missing values Coefficients Term Coef SE Coef T P 95% CI Constant 7.76751 3.38499 2.29469 0.027 (0.926194, 14.6088) AdvExp (thousands) 0.11289 0.02793 4.04219 0.000 (0.056446, 0.1693) Summary of Model S = 9.67521 R-Sq = 29.00% R-Sq(adj) = 27.23% PRESS = 4199.82 R-Sq(pred) = 20.37% Analysis of Variance Source DF Seq SS Adj SS Adj MS F P Regression 1 1529.52 1529.52 1529.52 16.3393 0.0002341 AdvExp (thousands) 1 1529.52 1529.52 1529.52 16.3393 0.0002341 Error 40 3744.39 3744.39 93.61 Total 41 5273.90 General Regression Analysis: New WC Cases versus AdvExp (thousands) Regression Equation New WC Cases = 24.5741 + 0. AdvExp (thousands) 42 cases used, 6 cases contain missing values Coefficients Term Coef SE Coef T P 95% CI Constant 24.5741 3.36671 7.29915 0.000 (17.7697, 31.3785) AdvExp (thousands) Summary of Model 0.0098 0.02778 0.35370 0.725 (-0.0463, 0.0660) S = 9.62296 R-Sq = 0.31% R-Sq(adj) = -2.18% PRESS = 4135.49 R-Sq(pred) = -11.30% Analysis of Variance Source DF Seq SS Adj SS Adj MS F P Regression 1 11.58 11.58 11.5848 0.125104 0.725421 AdvExp (thousands) 1 11.58 11.58 11.5848 0.125104 0.725421 Error 40 3704.06 3704.06 92.6015 Total 41 3715.64 Regression Analysis: New PI Cases versus AdvExp (thousands) The regression equation is New PI Cases = 7.768 + 0.1129 AdvExp (thousands) S = 9.67521 R-Sq = 29.0% R-Sq(adj) = 27.2% Analysis of Variance Source DF SS MS F P Regression 1 1529.52 1529.52 16.34 0.000 Error 40 3744.39 93.61 Total 41 5273.90 Fitted Line: New PI Cases versus AdvExp (thousands) Regression Analysis: New WC Cases versus AdvExp (thousands) The regression equation is New WC Cases = 24.57 + 0.00982 AdvExp (thousands) S = 9.62296 R-Sq = 0.3% R-Sq(adj) = 0.0% Analysis of Variance Source DF SS MS F P Regression 1 11.58 11.5848 0.13 0.725 Error 40 3704.06 92.6015 Total 41 3715.64 Fitted Line: New WC Cases versus AdvExp (thousands) For the new PI cases, assume an advertising expenditure of x = $150,000. Use Minitab (setting x = 150) to find the 95% Confidence Interval and the 95% Prediction Interval. Use the Minitab Tutorials for the Minitab directions. Then, in your post, interpret the output for the 95% Confidence Interval and the 95% Prediction Interval. What is the difference conceptually between the two intervals. Following is the relevant section of the Week 6 Lecture. It provides focus for the needed analysis of inference in regression. NOTE: in this Discussion topic we will focus on inference about the population values of the dependent variable, for a given value of the independent variable. ALSO NOTE: the Week 6 Lecture demonstrates how to do regression for the New PI Cases. For our Discussion, you are to apply the same ideas and tools to the New WC Cases. Also, see the text book Section 10.6 "Using the Model for Estimation and Prediction" pp. 599-604. Specifically, for this Week's Case, see p. 604. Here is the relevant portion of the Week 6 Lecture. I've put the relevant portion of the Lecture in Bold. • In the following regression output, we will begin by reviewing on the Minitab Regression output the descriptive statistics that we observed on the scatterplot. These are highlighted in yellow, with comments. o It is important to note at the outset that we will be making inferences of two kinds. ▪ Inferences about the population regression coefficient. These inferences will have two parts. ▪ Hypothesis tests about the population regression coefficient ▪ Confidence intervals about the population regression coefficient ▪ Predictions about the dependent variable for a given value of the independent variable. These predictions will be of two kinds. ▪ For a given x, across many instances of the same value of x, the predicted mean of the dependent variable (this is called a "confidence interval"). ▪ For a given x, for a single instance of the given value of x, the predicted value of the dependent variable (this is called a "prediction interval"). A confidence interval is calculated from the sample data and is a range estimates for the population parameter (i.e. the population mean). The prediction interval provides a range of values given a single observation. For the legal advertising case, when the advertising expenditure is $150,000, the confidence intervals are as follows: 95% Confidence Interval: (20.89, 28.51) This indicates that for all months with a cumulative advertising expenditure of $150,000 we can be 95% confident that the average number of personal injury cases for each month will be between 21 and 28 cases. 95% Prediction Interval: (4.78, 44.62) This indicates that we can be 95% confident that, for a given month with a 6-month advertising expenditure of $150,000, the law firm will add between 5 and 44 new personal injury cases. General Regression Analysis: New PI Cases versus AdvExp (thousands) Regression Equation New PI Cases = 7.76751 + 0.112891 AdvExp (thousands) 42 cases used, 6 cases contain missing values Coefficients Term Coef SE Coef T P 95% CI Constant 7.76751 3.38499 2.29469 0.027 (0.926194, 14.6088) AdvExp (thousands) 0.11289 0.02793 4.04219 0.000 (0.056446, 0.1693) Summary of Model S = 9.67521 R-Sq = 29.00% R-Sq(adj) = 27.23% PRESS = 4199.82 R-Sq(pred) = 20.37% Analysis of Variance Source DF Seq SS Adj SS Adj MS F P Regression 1 1529.52 1529.52 1529.52 16.3393 0.0002341 AdvExp (thousands) 1 1529.52 1529.52 1529.52 16.3393 0.0002341 Error 40 3744.39 3744.39 93.61 Total 41 5273.90 Predicted Values for New Observations New Obs Fit SE Fit 95% CI 95% PI 1 24.7011 1.88524 (20.8909, 28.5113) (4.77904, 44.6232) Values of Predictors for New Observations AdvExp New Obs (thousands) 1 150