Question

Number of Certified Organic Farms in the United States, 2001–2008
Year	Farms
2001	6,375
2002	6,730
2003	7,441
2004	7,425
2005	7,882
2006	8,758
2007	10,297
2008	12,019

(a) Use Excel, MegaStat, or MINITAB to fit three trends (linear, quadratic, exponential) to the time series. (A negative value should be indicated by a minus sign. Do not round the intermediate calculations. Round your final answers to 2 decimal places.)

Linear	yt = ____ xt + ______
Quadratic	yt = ____ xt2 +_____ xt + _____
Exponential	yt = _____ e ____x

(b) Use each of the three fitted trend equations to make numerical forecasts for the next 3 years. (Round the intermediate calculations to 2 decimal places and round your final answers to 1 decimal place.)

T Linear| Exponential | Quadratic

9 _________ _________ _________

10 _________ _________ _________

11 _________ ___________ _________

Answer 1

(a) Use Excel, MegaStat, or MINITAB to fit three trends (linear, quadratic, exponential) to the time series. (A negative value should be indicated by a minus sign. Do not round the intermediate calculations. Round your final answers to 2 decimal places.)

Linear	yt = 735.13 xt + 5057.79
Quadratic	yt = 122.93 xt2 +(-371.28 )xt + 6901.80
Exponential	yt = 5594.98 e 1.09x

(b) Use each of the three fitted trend equations to make numerical forecasts for the next 3 years. (Round the intermediate calculations to 2 decimal places and round your final answers to 1 decimal place.)

t	Linear	Exponential	Quadratic
9	11,673.96	11989.80	13,517.98
10	12,409.10	13049.42	15,482.46
11	13,144.23	14202.69	17,692.80

Mega stat used

Regression Analysis
	r²	0.878	n	8
	r	0.937	k	1
	Std. Error	724.021	Dep. Var.	farm
ANOVA table
Source	SS	df	MS	F	p-value
Regression	22,697,535.7202	1	22,697,535.7202	43.30	.0006
Residual	3,145,237.1548	6	524,206.1925
Total	25,842,772.8750	7
Regression output					confidence interval
variables	coefficients	std. error	t (df=6)	p-value	95% lower	95% upper
Intercept	5,057.7857	564.1525	8.965	.0001	3,677.3543	6,438.2172
t	735.1310	111.7188	6.580	.0006	461.7648	1,008.4971
Predicted values for: farm
		95% Confidence Intervals		95% Prediction Intervals
t	Predicted	lower	upper	lower	upper	Leverage
9	11,673.964	10,293.533	13,054.396	9,428.032	13,919.896	0.607
10	12,409.095	10,780.328	14,037.862	10,002.541	14,815.649	0.845
11	13,144.226	11,260.180	15,028.272	10,558.061	15,730.391	1.131
Regression Analysis
	R²	0.977
	Adjusted R²	0.967	n	8
	R	0.988	k	2
	Std. Error	348.216	Dep. Var.	farm
ANOVA table
Source	SS	df	MS	F	p-value
Regression	25,236,502.4405	2	12,618,251.2202	104.06	.0001
Residual	606,270.4345	5	121,254.0869
Total	25,842,772.8750	7
Regression output					confidence interval
variables	coefficients	std. error	t (df=5)	p-value	95% lower	95% upper
Intercept	6,901.8036	485.8111	14.207	3.11E-05	5,652.9864	8,150.6207
t	-371.2798	247.6869	-1.499	.1942	-1,007.9792	265.4197
t*t	122.9345	26.8654	4.576	.0060	53.8748	191.9943
Predicted values for: farm
			95% Confidence Intervals		95% Prediction Intervals
t	t*t	Predicted	lower	upper	lower	upper
9	81	13,517.982	12,269.165	14,766.799	11,981.500	15,054.465
10	100	15,482.458	13,569.865	17,395.052	13,370.765	17,594.151
11	121	17,692.804	14,966.036	20,419.571	14,822.875	20,562.733

For exponential, log transform farm values are used. Results are back transformed.

Regression Analysis
	r²	0.925	n	8
	r	0.962	k	1
	Std. Error	0.064	Dep. Var.	log.form
ANOVA table
Source	SS	df	MS	F	p-value
Regression	0.3012	1	0.3012	74.03	.0001
Residual	0.0244	6	0.0041
Total	0.3256	7
Regression output					confidence interval
variables	coefficients	std. error	t (df=6)	p-value	95% lower	95% upper
Intercept	8.6296	0.0497	173.617	2.46E-12	8.5080	8.7512
t	0.0847	0.0098	8.604	.0001	0.0606	0.1088
Predicted values for: log.form
		95% Confidence Intervals		95% Prediction Intervals
t	Predicted	lower	upper	lower	upper	Leverage
9	9.3918117	9.2701884	9.5134350	9.1939332	9.5896902	0.607
10	9.4764991	9.3329961	9.6200020	9.2644689	9.6885292	0.845
11	9.5611864	9.3951919	9.7271808	9.3333316	9.7890412	1.131
Predicted and transformed	11989.8019
	13049.4212
	14202.6861

Transformed coefficients:

5594.984

1.088