Bill McFarland is a real estate broker who specializes in selling farmland in a large western state.

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Question:

Bill McFarland is a real estate broker who specializes in selling farmland in a large western state. Because Bill advises many of his clients about pricing their land, he is interested in developing a pricing formula of some type. He feels he could increase his business significantly if he could accurately determine the value of a farmer’s land. A geologist tells Bill that the soil and rock characteristics in most of the area that Bill sells do not vary much. Thus the price of land should depend greatly on acreage. Bill selects a sample of 30 plots recently sold. The data is found below (in Minitab), where X=Acreage and Y=Price ($1,000s).

PRICE ACREAGE PREDICT
60 20.0 50
130 40.5 250
25 10.2
300 100.0
85 30.0
182 56.5
115 41.0
24 10.0
60 18.5
92 30.0
77 25.6
122 42.0
41 14.0
200 70.0
42 13.0
60 21.6
20 6.5
145 45.0
61 19.2
235 80.0
250 90.0
278 95.0
118 41.0
46 14.0
69 22.0
220 81.5
235 78.0
50 16.0
25 10.0
290 100.0

Correlations: PRICE, ACREAGE

Pearson correlation of PRICE and ACREAGE = 0.997
P-Value = 0.000

Regression Analysis: PRICE versus ACREAGE

The regression equation is
PRICE = 2.26 + 2.89 ACREAGE

Predictor Coef SE Coef T P
Constant 2.257 2.231 1.01 0.320
ACREAGE 2.89202 0.04353 66.44 0.000

S = 7.21461 R-Sq = 99.4% R-Sq(adj) = 99.3%

Analysis of Variance

Source DF SS MS F P
Regression 1 229757 229757 4414.11 0.000
Residual Error 28 1457 52
Total 29 231215

Predicted Values for New Observations

New Obs Fit SE Fit 95% CI 95% PI
1 146.86 1.37 (144.05, 149.66) (131.82, 161.90)
2 725.26 9.18 (706.46, 744.06) (701.35, 749.17)XX

XX denotes a point that is an extreme outlier in the predictors.

Values of Predictors for New Observations

New Obs ACREAGE
1 50
2 250

a. Analyze the above output to determine the regression equation.
b. Find and interpret in the context of this problem.
c. Find and interpret the coefficient of determination (r-squared).
d. Find and interpret coefficient of correlation.
e. Does the data provide significant evidence (= .05) that the acreage can be used to predict the price? Test the utility of this model using a two-tailed test. Find the observed p-value and interpret.
f. Find the 95% confidence interval for mean price of plots of farmland that are 50 acres. Interpret this interval.
g. Find the 95% prediction interval for the price of a single plot of farmland that is 50 acres. Interpret this interval.
h. What can we say about the price for a plot of farmland that is 250 acres?