ADM2304X Assignment#4 Spring-Summer 2017 ADM2304X: Assignment#4 (100 Marks) Prof.: Dr. Suren Phansalker Due Date: Must be Uploaded to Brightspace in 'pdf' Format by Sunday, July 23rd, 2017, before 23:30 hrs. Late upload with severe penalty is possible before 23:30 on Monday, July 24th. Integrity Statement: Must be Attached and Signed/Printed. General Instructions: When you perform a test of hypothesis, you must include: i. the \"Null\" and \"Alternate\" hypotheses, ii. calculated value of the test statistic, iii. the level of significance and the critical value of the statistic, iv. your decision rule and the conclusion reached to reject or not reject the null hypothesis. If you use p-value, you must calculate it first, and relate the p-value to the level of significance in reaching your conclusion. You are welcome to use the \"CDF\" and "INVCDF" functions in MiniTab for accuracy, if you find the values in Statistical Tables are inadequate. MiniTab can always be used to do intermediate calculations of various \"sums of squares\" 'SS' , \"the Standard Deviations and Standard Errors\" 's', 'SE', and also simply to double-check your manual calculations. If nothing is specified, you may use MiniTab to perform the hypothesis test. You must paste the relevant output into your assignment. This output replaces only the manual computation of the 'SS', 's', 'SE', the test statistic, p-value or the confidence interval. You must supply all other parts, mentioned above, to make your testing procedure complete. P.S.: "Manually" implies doing it the way it would be done on paper and pencil. Qu.#1 (20 Marks) Leisure Activity Products Manufacturing Company (LAPMC) tracked its monthly 'Market Share in %' or 'Y' and its monthly 'Advertising Expenses' in $K (thousands of dollars) or 'X'. The following data was obtained over a period of time. Use this data for both Qu.#1 and Qu.#2. Data Display X Y 64 6 88 8 156 22 142 20 76 10 96 14 104 12 120 18 130 16 146 20 150 22 160 28 174 26 a. Draw the scatter plot with the regression line for the above data and find the 'correlation coefficient' for the same. Comment on both the nature of the scatter plot and the correlation coefficient. Is the correlation coefficient significant? Justify your answer numerically without doing any test. b. Obtain the linear regression equation for all the above data; call it the 'full' regression equation. c. Manually calculate the R2 and R2Adj for this 'full' regression equation and comment on the quality of the regression. Explain the significance of R2. What is the relation between the 'correlation coefficient' and 'coefficient of determination'? d. Plot the standardized residuals in 4 different ways and comment on at least 2 different plots. Justify the aptness of this 'full' regression equation by stating if the assumptions for this regression model have been met or not. e. The model you obtain in part 'b' above, may contain 'unusual' observation/s which is/are undesirable. Drop this/these undesirable observation/s and find the 'reduced' regression equation. Manually calculate the R2 and R2Adj for this 'reduced' regression equation and comment on the quality of the regression. Explain why there is a difference in the quality of this 'reduced' regression equation and the 'full' regression equation. Prof.: Dr. Suren Phansalker Management Science 1 University of Ottawa ADM2304X Assignment#4 Spring-Summer 2017 Qu.#2 (20 Marks) In this Question, use the 'full' regression model you obtained in Qu.#1, part 'b'. a. Manually conduct the Hypothesis Test on '1' at the 5% 'Level of Significance'. Use 'Critical Value' of 't' to reach your conclusion. Also calculate the p-Value and based on it, justify your conclusion. b. Manually conduct the Hypothesis Test on 'Significance of Regression' at the 5% 'Level of Significance'. Use 'Critical Value' of 'F' to reach your conclusion. Also calculate the p-Value and based on it, justify your conclusion. c. Manually obtain the 99% Confidence Interval for the 'Percentage Market Share' if the 'Monthly Ad Expense' is $170,000. What does this Interval mean? Explain in brief. d. Manually obtain the 99% Prediction Interval for the 'Percentage Market Share' if the 'Monthly Ad Expense' is $170,000. What does this Interval mean? Explain in brief. e. Why is the 'Prediction Interval' broader than the 'Confidence Interval'? Explain in brief. f. With the whole of the data or the thirteen ordered pairs, obtain a linear regression equation, by dropping the constant term. What will be its R2 and R2Adj ? Is dropping the constant term, particularly essential? Comment briefly. You will have to download either one of the MiniTab Files; 'Assign#4_MTB2304_SS17_V14.MTW' or 'Assign#4_MTB2304_SS17_V17.MTW' . One of these files (which ever that works for you) is needed to do your Qu.#3 and Qu.#4 in Assignment#4. Qu.#3 (30 Marks) Variable Definitions: SPrice: Selling Price in $K (thousands of dollars) SqFt: Area in Square Feet x100 NumFlrs: Number of Floors BdRms: Number of Bedrooms (can be fractional) Baths: Number of Bathrooms (can also be fractional) In a red-hot real estate market, a real estate company wanted to establish some type of a linear relationship between the dependent / response variable of 'Selling Price' of houses in $K and the 4 independent / predictor variables given above. The data is given in the file: 'Assign#4_MTB2304_SS17_V14.MTW' or 'Assign#4_MTB2304_SS17_V17.MTW'. a. Use each of the 4 independent variables, one at a time and obtain 4 Simple Linear Regression (SLR) Equations for the response variable, 'Selling Price'. Rank these 4 SLR equations in 'Descending Order' from the 'Best' to the 'Worst'. Explain what specific criteria you use in this ranking. b. Now use all the 4 independent variables to obtain 1 Full Multiple Linear Regression (MLR) equation for all the independent/explanatory variables. c. Manually calculate the R2 and R2Adj for this MLR equation and comment on the quality of the regression. Explain the significance of R2. d. By simply observing the MiniTab printout and without doing any formal analysis, comment on which of the parameters, 0, 1, 2, 3, 4 are significant. What specific numerical criterion did you use? Explain briefly. e. Now drop the worst performing independent variable and use the remaining three independent variables, to obtain the new 'Best' MLR equation. Prof.: Dr. Suren Phansalker Management Science 2 University of Ottawa ADM2304X Assignment#4 Spring-Summer 2017 Manually calculate the R2 and R2Adj for this new 'Best' MLR equation and comment on the quality of this regression. Use "Best Subset" approach to justify if this 'Best' Multiple Regression is in fact the 'Best'! g. For this 'Best' MLR equation, plot the 4 standardized residuals and comment on at least 2 different plots. Justify the aptness of this 'Best' MLR equation by stating if the assumptions for this regression model have been met or not. Do you see any possible problems? h. Explain why the 'Best' MLR equation is better than the MLR equation with 4 independent variables. i. In part 'b' above, you obtained your "Full" MLR equation with all the data. Now, drop the most 'unusual observation' (just one observation!) and find the new equation and specify what it would be. Is it better than the "Full" MLR equation? f. Qu.#4 (30 Marks) For this Question, use the 'Best' MLR equation that you have obtained in Qu.#3, part 'e'. a. Manually conduct the Hypothesis Test on '1' at the 5% 'Level of Significance'. Use 'Critical Value'of 't' to reach your conclusion. Also calculate the p-Value and based on it, justify your conclusion. b. Manually conduct the Hypothesis Test on 'Significance of Regression' at the 5% 'Level of Significance'. Use 'Critical Value' of 'F' to reach your conclusion. Also calculate the p-Value and based on it, justify your conclusion. c. Manually Obtain the 99% Confidence Interval for the 'Selling Price' of the house when the area is 4000 square feet, and there are 2 floors, 5 bedrooms and 3 bath rooms. What does this Interval mean? Explain in brief. d. Manually obtain the 99% Prediction Interval for the 'Selling Price' when the area is 4000 square feet, and there are 2 floors, 5 bedrooms and 3 bath rooms. What does this Interval mean? Explain in brief. e. In this 'Best' MLR, the VIF for predictor variable 'SqFt' is 1.879. Explain how this number is calculated and demonstrate with actual calculations. What is the significance of this VIF number? f. 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