Question

Managerial Report, Chapter 15 (modified) 1. Develop the following estimated regression equations, using Amount Charged as your Dependent variable: a. First using annual income as the Independent variable b. Second using household size as the Independent variable Which variable is the better predictor of annual credit card charges? To answer this question, provide and interpret your R2 for each model. 2. Develop an estimated regression equation with annual income and household size as the Independent variables. As your Dependent variable, use Amount Charged. How does the R2 change? Interpret this change. 3. What is the predicted annual credit card charge (Amount Charged) for a three-person household (3) with an annual income of $40,000 (be careful to notice the units in your model)? 4. Plot the standardized residuals (y-axis) and the predicted values (x-axis) for question 2. Based on what you see, what is your conclusion about the Variance of the error term (u)? It is homoscedastic (same variance) or heteroskedastic (not the same variance)? 5. Using your understanding of Hypothesis testing, provide the null (H0) and alternative (Ha) hypotheses for a hypothesis test using the Student-t distribution for an analysis of each Independent variable. Using the regression output, determine whether each Independent variable is statistically significant. Provide support for your answer. Assume adherence to the industry standard of =.05. Conduct these tests for questions 1 and 2 (so you will look at the Independent variables in 3 regression models).

Income ($1000s)	Household Size	Amount Charged ($)
54	3	4,016
30	2	3,159
32	4	5,100
50	5	4,742
31	2	1,864
55	2	4,070
37	1	2,731
40	2	3,348
66	4	4,764
51	3	4,110
25	3	4,208
48	4	4,219
27	1	2,477
33	2	2,514
65	3	4,214
63	4	4,965
42	6	4,412
21	2	2,448
44	1	2,995
37	5	4,171
62	6	5,678
21	3	3,623
55	7	5,301
42	2	3,020
41	7	4,828
54	6	5,573
30	1	2,583
48	2	3,866
34	5	3,586
67	4	5,037
50	2	3,605
67	5	5,345
55	6	5,370
52	2	3,890
62	3	4,705
64	2	4,157
22	3	3,579
29	4	3,890
39	2	2,972
35	1	3,121
39	4	4,183
54	3	3,730
23	6	4,127
27	2	2,921
26	7	4,603
61	2	4,273
30	2	3,067
22	4	3,074
46	5	4,820
66	4	5,149

Answer 1

1)

a)

Model 1

Regression Equation

Amount Charged = 2,204 + 40.48 * Income

R Square = 0.3981 or 39.81%

Coefficients Hypothesis testing

Income

Null and alternate Hypothesis

H₀: The coefficient of independent variable is 0

H_a: The coefficient of independent variable is not 0

Result

Since the p-value for the Coefficient is less than 0.05, we reject the null hypothesis ie coefficient of independent variable is not 0.

b)

Model 2

Regression Equation

Amount Charged = 2,581.94 + 404.13* Household Size

R Square = 0.5668 or 56.68%

Coefficients Hypothesis testing

Household Size

Null and alternate Hypothesis

H₀: The coefficient of independent variable is 0

H_a: The coefficient of independent variable is not 0

Result

Since the p-value for the Coefficient is less than 0.05, we reject the null hypothesis ie coefficient of independent variable is not 0.

Model 2 is better than Model 1 as the R Square for Model 2 is higher than that of Model 1 ie Model 2 explains 56.68% of variation in Amount Charged while Model 1 only explains 39.81%.

2)

Model 3

Regression Equation

Amount Charged = 1,304.90 + 33.13 * Income + 356.30* Household Size

R Square = 0.8256 or 82.56%

Coefficients Hypothesis testing

Income

Null and alternate Hypothesis

H₀: The coefficient of independent variable is 0

H_a: The coefficient of independent variable is not 0

Result

Since the p-value for the Coefficient is less than 0.05, we reject the null hypothesis ie coefficient of independent variable is not 0.

Coefficients Hypothesis testing

Household Size

Null and alternate Hypothesis

H₀: The coefficient of independent variable is 0

H_a: The coefficient of independent variable is not 0

Result

Since the p-value for the Coefficient is less than 0.05, we reject the null hypothesis ie coefficient of independent variable is not 0.

Out of the 3 Models, Model 3 is the best as R Square for Model 3 is highest ie 82.56% of the variation in the dependent Variable can be explained by the independent variables.

3)

When Household Size = 3 and Income = 40,

Using Model 3 we get,

Amount Charged = 1,304.90 + 33.13 * 40 + 356.30* 3 = 3699.113 $

4)

Since there is no definite pattern in the above plot hence, we can conclude that It is homoscedastic (same variance)

5)

Already answered in above parts.