Question

A horticulturist is studying the relationship between tomato plant height and fertilizer amount. Thirty tomato plants grown in similar conditions were subjected to various amounts of fertilizer over a four-month period, and then their heights were measured. The results are shown in the accompanying table. Fertilizer Amount (ounces) Tomato Plant Height (inches) 1.9 20.4 5.0 29.1 4.2 28.2 1.3 24.3 4.9 29.2 5.4 29.8 3.1 34.6 0.0 25.3 2.3 26.2 2.5 25.7 0.9 26.5 1.0 28.6 4.5 25.7 3.8 36.8 5.3 27.2 2.3 33.4 4.0 25.1 1.4 22.1 3.9 40.6 4.0 24.5 3.6 39.7 0.6 21.1 1.6 25.4 2.5 29.6 4.5 27.6 3.6 32.6 2.0 33.5 0.0 22.5 2.5 27.6 3.1 36.4 Picture Click here for the Excel Data File a. Estimate the linear regression model: Height = β0 + β1Fertilizer + ε. (Round your answers to 3 decimal places.) Heightˆ = 25.000 + 1.290 Fertilizer b-1. Estimate the quadratic regression model: Height = β0 + β1Fertilizer + β2Fertilizer2 + ε. (Round your answers to 3 decimal places.) Heightˆ = 21.100 + 5.460 Fertilizer + 0.759 Fertilizer2 b-2. Using the quadratic results, find the fertilizer amount at which the height reaches a minimum or maximum. (Round your intermediate values and final answer to 3 decimal places.) Fertilizer amount = 3.9 ounces c. Use the best-fitting model to predict, after a four-month period, the height of a tomato plant that received 3.0 ounces of fertilizer. (Round your intermediate values and final answer to 2 decimal places.) Heightˆ = 30.649 inches

Answer 1

(a) Height^=24.956+1.291*Fertilizer

following regression analysis information has been generated using ms-excel

SUMMARY OUTPUT
Regression Statistics
Multiple R	0.386414872
R Square	0.149316453
Adjusted R Square	0.118934898
Standard Error	4.93026189
Observations	30
ANOVA
	df	SS	MS	F	Significance F
Regression	1	119.4641621	119.4642	4.914707	0.034921943
Residual	28	680.6095046	24.30748
Total	29	800.0736667
	Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%
Intercept	24.95578271	1.891309847	13.19497	1.54E-13	21.08161016	28.82995525
X Variable 1	1.290857863	0.582276943	2.216914	0.034922	0.098117631	2.483598095

(b) Height^=21.0677+5.456*fertilizer-0.759*fertilizer²

following regression analysis information has been generated using ms-excel

SUMMARY OUTPUT
Regression Statistics
Multiple R	0.517861295
R Square	0.268180321
Adjusted R Square	0.213971456
Standard Error	4.656773856
Observations	30
ANOVA
	df	SS	MS	F	Significance F
Regression	2	214.5640126	107.282	4.947167	0.014772539
Residual	27	585.5096541	21.68554
Total	29	800.0736667
	Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%
Intercept	21.06783451	2.576457603	8.177055	8.8E-09	15.78138023	26.35428878
X Variable 1	5.455905881	2.063550349	2.643941	0.013482	1.221850351	9.689961412
X Variable 2	-0.75923721	0.362554012	-2.09414	0.045769	-1.50313659	-0.01533783

(b2) answer is fertilizer=3.594

the quadratice line is given as , Height^=21.0677+5.456*fertilizer-0.759*fertilizer²

using maximization and minimization problem, we differenitate Height with respect ot fertilizer and putting them 0

Height/fertilizer=5.456-2*0.759fertilizer

and putting this equation equal to zero and we get

fertilizer=5.456/(2*0.759)=3.594

since ²Height/fertilizer² is negative , so at fertilizer=3.594, height will be maximum

(c) for fertilizer=3, the height=21.0677+5.456*3-0.759*3*3=30.60

here R-square for quadratic equation/model is more than linear equation/model , so we used quadratic model for the prediction.