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Week 3 Case Study (Case Study #3)^[i]

Based on your Case Study #1 and Case Study #2 analyses, the QuickFix general manager is not sure that the model that you developed will meet his needs. He wants you to give him more information so he can decide whether the predictions will be helpful to him. To address his request, you decide to calculate confidence and prediction intervals and test some of the regression assumptions.

Part 1

In Part 1 of this case study, you will calculate a predicted number of vehicles served using a given number of bays and a given population using the Bays and Population model from Case Study #1. You will also need to calculate a modified regression model to find the confidence interval around the predicted value. You will use the Bays and Populationworksheet in the QuickFix Vehicles Case Study Data.xlsx workbook for this part of the case study.

1. Use the Bays and Population model from Case Study #1 to predict the number of vehicles served for a location where Bays is 4 and Population is 50,000. Remember that the data uses population in thousands to develop the model so you will need to use 50 instead of 50,000 in the equation to calculate the predicted value. Write your answer in the box below.

1. Modified Multiple Regression for Interval Estimation.
   1. Run a modified multiple regression model using Bays and Population to construct a confidence interval and prediction interval for a location where Bays is 4 and Population is 50,000. If you are unsure how to run a modified regression model, you can check section 15.3 of the textbook or rewatch the video in Canvas covering this topic. Label your results in an Excel workbook using the prompt number.
   2. Using the modified multiple regression model, construct the 95% confidence interval for the mean expected number of vehicles served for a location where Bays is 4 and Population is 50,000. You can use the interval estimation tool provided in Canvas to assist with calculation of the interval. Label your results in an Excel workbook using the prompt number. Also, write your answer in the box below.

1. Using the modified multiple regression model, construct the 95% prediction interval for the expected number of vehicles served for a location where Bays is 4 and Population is 50,000. You can use the interval estimation tool provided in Canvas to assist with calculation of the interval. Label your results in an Excel workbook using the prompt number. Also, write your answer in the box below.

Part 2

In Part 2 of this case study, you will test whether the Bays and Population model from Case Study #1 model meets some specific regression assumptions. You will use the output from the Bays and Population model from Case Study #1 to answer question for Part 2. You will specifically need to review the residual plots and line fit plots. You will also need to run correlation matrices to test some assumptions. You will use the Bays and Population worksheet in the QuickFix Vehicles Case Study Data.xlsx workbook for this part of the case study.

1. Tests of Assumptions
   1. Using the Bays and Population model from Case Study #1, indicate whether the Linearity assumption (Assumption 1) is met and provide an explanation of how you reached your conclusion. See the video in Canvas on how to test assumptions. Label your results in an Excel workbook using the prompt number and write your explanation in the box below.

1. Using the Bays and Population model from Case Study #1, indicate whether the Multicollinearity assumption (Assumption 2) is met and provide an explanation of how you reached your conclusion. See the video in Canvas on how to test assumptions. Label your results in an Excel workbook using the prompt number and write your explanation in the box below.

1. Using the Bays and Population model from Case Study #1, indicate whether the Equal Variance assumption (Assumption 3) is met and provide an explanation of how you reached your conclusion. See the video on to test assumptions. Label your results in an Excel workbook using the prompt number and write your explanation in the box below.

1. Using the Bays and Population model from Case Study #1, indicate whether the Endogeneity assumption (Assumption 5) is met and provide an explanation of how you reached your conclusion. See the video in Canvas on how to test assumptions. Label your results in an Excel workbook using the prompt number and write your explanation in the box below.

^[i] This case study is adapted from Exercises 17.1, problem 16, page 598 of Business Statistics: communicating with numbers, Jaggia and Kelly, Fourth Edition. EXCEL DATA:

Vehicles Served	Bays	Population in Thousands
200	3	15
351	3	22
382	3	35
294	3	52
223	3	47
309	3	26
302	3	45
369	3	25
312	3	16
289	3	10
304	3	11
233	3	15
313	3	48
285	3	51
298	3	16
224	3	34
403	3	22
282	3	12
299	3	36
200	3	15
366	3	22
385	3	35
291	3	52
238	3	47
308	3	26
289	3	45
368	3	25
312	3	16
292	3	10
306	3	11
226	3	15
301	3	48
278	3	51
283	3	16
233	3	34
404	3	22
278	3	12
301	3	36
214	4	30
250	4	37
288	4	42
352	4	45
345	4	48
410	4	62
259	4	63
331	4	54
401	4	29
425	4	37
428	4	58
407	4	19
340	4	50
340	4	38
328	4	41
427	4	51
330	4	29
410	4	42
339	4	57
427	4	60
403	4	24
216	4	30
254	4	37
289	4	42
359	4	45
347	4	48
399	4	62
245	4	63
316	4	54
394	4	29
421	4	37
438	4	58
410	4	19
339	4	50
355	4	38
314	4	41
433	4	51
315	4	29
396	4	42
332	4	57
437	4	60
392	4	24
325	5	25
317	5	29
344	5	36
376	5	39
369	5	44
494	5	72
377	5	26
273	5	66
273	5	63
436	5	25
377	5	34
358	5	65
355	5	32
370	5	71
357	5	69
357	5	25
353	5	27
293	5	35
366	5	28
373	5	62
457	5	42
317	5	25
316	5	29
345	5	36
379	5	39
376	5	44
498	5	72
369	5	26
287	5	66
284	5	63
440	5	25
372	5	34
373	5	65
366	5	32
368	5	71
346	5	69
359	5	25
356	5	27
282	5	35
363	5	28
372	5	62
448	5	42
318	6	49
354	6	54
512	6	77
464	6	74
402	6	50
468	6	66
485	6	64
400	6	47
380	6	57
397	6	38
394	6	37
321	6	40
395	6	44
378	6	76
459	6	62
392	6	36
393	6	36
380	6	60
397	6	44
318	6	49
363	6	54
513	6	77
453	6	74
387	6	50
480	6	66
475	6	64
391	6	47
374	6	57
382	6	38
380	6	37
323	6	40
389	6	44
382	6	76
463	6	62
394	6	36
403	6	36
374	6	60
385	6	44
495	7	56
325	7	57
509	7	93
491	7	86
520	7	57
336	7	79
328	7	86
416	7	85
508	7	67
332	7	46
432	7	84
430	7	63
411	7	51
356	7	72
503	7	81
416	7	74
335	7	45
408	7	46
418	7	45
416	7	58
509	7	56
330	7	57
523	7	93
506	7	86
535	7	57
333	7	79
318	7	86
412	7	85
518	7	67
330	7	46
446	7	84
432	7	63
420	7	51
347	7	72
488	7	81
421	7	74
350	7	45
414	7	46
416	7	45
430	7	58

CASE STUDY 1:

Week 1 Case Study (Case Study #1)^[i]

The new general manager of QuickFix, a chain of quick service, no appointment auto repair shops, wants to predict monthly vehicles served at each of his locations to allow him to better stock each location with the necessary parts, engine oil and other supplies to meet customer needs. He hires you to develop a statistical model. He provides you with a data file with the monthly number of vehicles served, number of garage bays and population within a 5-mile radius for 200 shop locations. You will use the Bays and Population worksheet in the QuickFix Vehicles Case Study Data.xlsxworkbook for this case study. To develop the model, you decide to perform the following steps.

1. Simple Regression for Bays
   1. Run a simple regression model using Bays to predict Vehicles Served. Label your results in an Excel workbook using the prompt number. Make sure to select all of the check box options at the bottom the regression tool pop-up. You will use them in Case Study 3.
   2. Write the regression equation for the Bays model using the variable names, intercept coefficient, and slope coefficient from the regression output. Write your answer in the box below.

Vehicles Served = 29.40561 * Bays + 220.8869

1. Interpret the slope coefficient for the Bays model Write your answer in the box below.

For each additionalbay, the average number ofVehicles Served increases by approximately 29.41 vehicles per month

1. Simple Regression for Population
   1. Run a simple regression model using Population to predict Vehicles Served. Label your results in an Excel workbook using the prompt number. Make sure to select all of the check box options at the bottom the regression tool pop-up. You will use them in Case Study 3.
   2. Write the regression equation for the Population model using the variable names, intercept coefficient, and slope coefficient from the regression output. Write your answer in the box below.

Vehicles Served =285.40 + 1.75 * Population

1. Interpret the slope coefficient for the Population model Write your answer in the box below.

For every additional1,000 peoplein the surrounding population, the average number ofVehicles Served increases by approximately 1.75 vehicles per month

1. Multiple Regression for Bays and Population
   1. Run a multiple regression using both Bays and Population. Label your results in an Excel workbook using the prompt number. Make sure to select all of the check box options at the bottom the regression tool pop-up. You will use them in Case Study 3.
   2. Write the regression equation for the Bays and Population model using the variable names, intercept coefficient, and slope coefficients from the regression output. Write your answer in the box below.

Vehicles Served =217.92 + 23.40 * Bays + 0.70 * Population

1. Interpret the slope coefficients for the model. Write your answer in the box below.

Bays:For each additionalbay, the average number ofVehicles Served increases by approximately 23.40 vehicles per month.

Population:For every additional1,000 people, the average number ofVehicles Served increases by approximately 0.70 vehicles per month

1. Select the best fitting model and provide an explanation of how you reached your conclusion including the measure of goodness-of-fit that you used. Write your answer in the box below.

Themultiple regression modelis the best fitting model. It has thehighest R?? value of 0.3532, which means it explainsmore variation in Vehicles Servedthan either of the simple regression models.

1. Write a sentence stating the percent of variation in Vehicles Served that is explained by the explanatory variables in the best fitting model. Write your answer in the box below.

The best fitting model explains35.32%of the variation inVehicles Servedusing the explanatory variablesBays and Population.

^[i] This case study is adapted from Exercises 17.1, problem 16, page 598 of Business Statistics: communicating with numbers, Jaggia and Kelly, Fourth Edition.