Question

Hog Wild Incorporated produces clean start, a brand of liquid laundry detergent. The company wishes to study the factors affecting the quarterly demand for the large size bottle of clean start in its sales regions. To do this the company has selected the following variables:

y = The demand for the large size bottle

X1 = The price of clean start

X2 = the average price of competitors with similar detergents

X3 = The advertising expenditures to promote clean start

X1	X2	X3	Y
3.85	3.80	5.50	7.38
3.75	4.00	6.75	8.51
3.70	4.30	7.25	9.52
3.70	3.70	5.50	7.50
3.60	3.85	7.00	9.33
3.60	3.80	6.50	8.28
3.60	3.75	6.75	8.75
3.80	3.85	5.25	7.87
3.80	3.65	5.25	7.10
3.85	4.00	6.00	8.00
3.90	4.10	6.50	7.89
3.90	4.00	6.25	8.15
3.70	4.10	7.00	9.10
3.75	4.20	6.90	8.86

1. Write down the model

2. Write down the prediction equation

3. S²⁼

4.Write down the 95% CI for B2

5. Write down the 95% CI for the E(Y) when:

X1=3.75, X2+4.20, X3=6.90

Answer 1

Objective: To study the factors affecting the quarterly demand for the large size bottle of clean start in its sales regions

In order to attain the objective, we must first establish a causal relationship between the dependent variable - Demand for the large size bottle using the predictors, Price of clean start, Average price of competitors with similar detergents and Advertising expenditures to promote clean start.

1. Assuming a linear relationship here, we may run a linear regression model, by regressing Demand for the large size bottle on the predictors, Price of clean start, Average price of competitors with similar detergents and Advertising expenditures to promote clean start to produce a regression model of the form:

where, and denote the intercept and the three slope coefficients respectively and is the residual.

Using excel,

We get the output:

2. We get the fitted regression equation as:

3. If by S, we mean Standard error,

Then S² = (0.252)² = 0.064

4. The 95% CI for slope is constructed using the formula:

From the output,

95% CI for slope corresponding to X2

= (-0.061, 3.325)

5. For X1=3.75, X2+4.20, X3=6.90

= 9.0043..................(subject to rounding error of coefficients)

Since, constructing prediction interval for a multiple regression model is a bit tedious, we may create the same using a software, here, using Rstudio:

The 95% Prediction interval is obtained as:

(8.374479,9.627679)