Question

The ski season started in Mount Sunapee! The ski and snowboard rental shop is located in the ski lodge area and offers a large selection of current snowboards, shaped skis, and helmets for children, women, and men. The winter rental shop is open every day that Mount Sunapee is open for skiing. The rental store maintains a current inventory of quality rental equipment with all rental equipment inventory replaced on a normal rotation basis every three to four years.

The owner of the store wants to understand better the factors that affect the demand for his ski and snowboard rental shop, so he can plan his capacity better. In particular, he wants to know how different factors, as the day of month, weekends, weather or school breaks affect their demand.

You will find data on number of rentals per day for December 2015 in this spreadsheet:

Date	Day	Weekend	Weather (good snow)	School Break	Shaped Ski Rentals	Snowboard Rentals	Helmet Rentals
01/12/2015	1	0	0	0	25	8	15
02/12/2015	2	0	0	0	22	5	10
03/12/2015	3	0	0	0	26	7	15
04/12/2015	4	0	0	0	35	10	20
05/12/2015	5	1	0	0	90	25	65
06/12/2015	6	1	0	0	85	20	62
07/12/2015	7	0	0	0	28	6	15
08/12/2015	8	0	0	0	22	5	12
09/12/2015	9	0	0	0	25	5	15
10/12/2015	10	0	0	0	28	7	17
11/12/2015	11	0	0	0	35	8	20
12/12/2015	12	1	1	0	120	35	85
13/12/2015	13	1	1	0	105	28	95
14/12/2015	14	0	1	0	48	15	32
15/12/2015	15	0	0	0	25	5	15
16/12/2015	16	0	0	0	35	8	21
17/12/2015	17	0	1	0	45	12	28
18/12/2015	18	0	1	0	60	15	37
19/12/2015	19	1	1	0	130	42	90
20/12/2015	20	1	1	0	110	32	80
21/12/2015	21	0	1	0	70	15	39
22/12/2015	22	0	1	0	53	7	30
23/12/2015	23	0	1	1	85	18	49
24/12/2015	24	0	1	1	90	22	55
25/12/2015	25	0	1	1	110	35	70
26/12/2015	26	1	1	1	170	45	105
27/12/2015	27	1	1	1	190	50	120
28/12/2015	28	0	1	1	110	42	75
29/12/2015	29	0	1	1	90	35	65
30/12/2015	30	0	1	1	75	28	50

Part 1

Let’s start creating a linear regression model of daily shaped ski rentals as a function of the day of month only. Are the regression coefficients statistically significant at p=0.01?

Select the best answer

1.Neither β0 nor β1 are statistically significant at p=0.01.

2.Both β0 and β1 are statistically significant at p=0.01.

3.Only β0 is statistically significant at p=0.01.

4.Only β1 is statistically significant at p=0.01.

Part 2

The owner of the winter rental shop wants you to analyze how weekends affect his daily demand for shaped ski rentals, since he has noticed that more people rent shaped skis on weekend days (e.g. Saturday and Sunday) than on weekdays (e.g. Monday through Friday). Create a regression model of shaped skis daily rentals as a function of weekends. Use as only independent variable a dummy variable with a value of 1 for weekend days and 0 for the weekdays.

What is the ?2 value for this model?

Part 3

Based on your previous experience working on another ski rental shop located in Vermont (US), you believe that school breaks affect the daily rental demand for shaped skis. You also think that a better model could be obtained by using a multiple regression approach. Your proposal is to analyze three different models to predict shaped ski rentals using two independent variables in each model: (A) one model using the day of month and weekend as the independent variables, (B) another using the day of month and school break as independent variables, and (C) a last one using weekend and school break as independent variables. Which of these models is the best considering predictive power and statistical significance of coefficients?

Select the best answer

1.The model using the day of month and weekend is the best

2.The model using the day of month and school break is the best

3.The model using weekend and school break is the best

4.The model using only weekend (from Part 2) is better than any of these

Part 4

Create now a multivariate regression model using three independent variables: weekend, school break, and weather as predictors of the shaped ski rentals. For weather use a dummy variable with a value of 1 for those days that the quality of snow were good.

What is the adjusted ?2 value for this model?

Part 5

According to the previous (Part 4) multivariate regression model that you have created using three independent variables: weekend, school break, and weather as predictors of the shaped ski rentals, which is the value of the intercept?

Answer 1

For the given data, using excel --> Data Analysis Tool:

Part 1: Creating a linear regression model of daily shaped ski rentals as a function of the day of month only:

From the output obtained, looking at the p-value of the coefficients obtained, we find that the intercept is not significant. However, the slope coefficient of the independent variable 'Day' is significant (p value = 0.000) at 0.01 level.

The best answer would be:

Only β1 is statistically significant at p=0.01.

Part 2

Creating a regression model of shaped skis daily rentals as a function of weekends.

From the output obtained, the ?2 value for this model is 0.527 (52.7%).

Part 3 Analyzing three different models to predict shaped ski rentals using two independent variables in each model: The model with the highest R2 ,(in fact Adj. R2), significant model (F statistic), with significant intercept and slope coefficients is considered the best model.

(A) Model using the day of month and weekend as the independent variables,

Model A:

R2 = 0.887 Significant: Model and Slope Coefficients

(B) Model using the day of month and school break as independent variables,

Model B:

R2 = 0.421 Significant: Model

(C) Model using weekend and school break as independent variables.ese

Model C:

R2 = 0.896 Significant: Model, Intercept and Slope coefficients

For Model in Part 2 and Models A,B and C in Part 3, comparing the R2 and significance values, we find that Model using weekend and school break is the best, considering predictive power and statistical significance of coefficients.

Part 4

Creating a multivariate regression model using three independent variables: weekend, school break, and weather as predictors of the shaped ski rentals:

The adjusted ?2 value for this model is obtained as 0.944. It implies that the variation in predictors 'School Break', ' Weekend' and Weather' together explains about 94.4% of the variation in the dependent variable - shaped ski rentals. This would be considered the best model so far.

Part 5

The value of the intercept in the above model is obtained as 26.582 (p-value = 0.000<0.01)