Question

1. Suppose that beer industry representatives hypothesize that the marginal propensity to buy beer out of an additional dollar of income is $0.01. Conduct a hypothesis test to determine if there is any validity to their conjecture. Use a = 1%.  
2. Conduct a hypothesis test to determine if the regressors are jointly significant in explaining the quantity of beer purchased. Be sure to state the null and alternative hypotheses formally

obs	q	pB	pL	pR	m
1	81.7	1.78	6.95	1.11	25088
2	56.9	2.27	7.32	0.67	26561
3	64.1	2.21	6.96	0.83	25510
4	65.4	2.15	7.18	0.75	27158
5	64.1	2.26	7.46	1.06	27162
6	58.1	2.49	7.47	1.1	27583
7	61.7	2.52	7.88	1.09	28235
8	65.3	2.46	7.88	1.18	29413
9	57.8	2.54	7.97	0.88	28713
10	63.5	2.72	7.96	1.3	30000
11	65.9	2.6	8.09	1.17	30533
12	48.3	2.87	8.24	0.94	30373
13	55.6	3	7.96	0.91	31107
14	47.9	3.23	8.34	1.1	31126
15	57	3.11	8.1	1.5	32506
16	51.6	3.11	8.43	1.17	32408
17	54.2	3.09	8.72	1.18	33423
18	51.7	3.34	8.87	1.37	33904
19	55.9	3.31	8.82	1.52	34528
20	52.1	3.42	8.59	1.15	36019
21	52.5	3.61	8.83	1.39	34807
22	44.3	3.55	8.86	1.6	35943
23	57.7	3.72	8.97	1.73	37323
24	51.6	3.72	9.13	1.35	36682
25	53.8	3.7	8.98	1.37	38054
26	50	3.81	9.25	1.41	36707
27	46.3	3.86	9.33	1.62	38411
28	46.8	3.99	9.47	1.69	38823
29	51.7	3.89	9.49	1.71	38361
30	49.9	4.07	9.52	1.69	41593

Answer 1

(a) The hypothesis being tested is:

H₀: β₄ = 0

H₁: β₄ ≠ 0

The p-value from the output is 0.0165.

Since the p-value (0.0165) is less than the significance level (0.05), we can reject the null hypothesis.

Therefore, we can conclude that the marginal propensity to buy beer out of an additional dollar of income is $0.01.

(b) The hypothesis being tested is:

H₀: β₁ = β₂ = β₃ = β₄ = 0

H₁: At least one β_i ≠ 0

R²	0.822
	Adjusted R²	0.794
	R	0.907
	Std. Error	3.569
	n	30
	k	4
	Dep. Var.	q
ANOVA table
Source	SS	df	MS	F	p-value
Regression	1,471.9316	4	367.9829	28.89	4.77E-09
Residual	318.4831	25	12.7393
Total	1,790.4147	29
Regression output					confidence interval
variables	coefficients	std. error	t (df=25)	p-value	95% lower	95% upper
Intercept	82.1587
pB	-23.7426	5.4294	-4.373	.0002	-34.9247	-12.5605
pL	-4.0774	3.8905	-1.048	.3046	-12.0900	3.9352
pR	12.9243	4.1639	3.104	.0047	4.3486	21.5000
m	0.0020	0.0008	2.571	.0165	0.0004	0.0036

The p-value is 0.0000.

Since the p-value (0.0000) is less than the significance level (0.05), we can reject the null hypothesis.

Therefore, we can conclude that the regressors are jointly significant in explaining the quantity of beer purchased.

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