Question

1. Post  one set of solutions with names of all participating students in the group to the Canvas course website.
2. The one file needs to be in either an MS Word format or PDF format
3. Graphs should be neat, clean and well-labeled.
4. “Explanations” and answers should given be given in the form of complete sentences.

                                                

One can determine how old a tree is by counting its rings, but that Requires cutting the tree down.  A forester wanted to investigate whether or not the age of a tree can be estimated simply from its diameter so they measured 27 trees of the same species that had been cut down, and counted the rings to determine the ages of the trees.  The following are the data collected.

Diameter in inches	Age in years	Diameter in inches	Age in years	Diameter in inches	Age in years	Diameter in inches	Age in years
3.8	4	6.8	12	9.3	23	11.5	34
3.8	5	6.9	13	9.9	25	11.7	35
5.7	8	7.6	14	9.8	28	11.7	38
5.0	8	7.6	16	10.6	29	11.7	38
5.8	8	8.1	18	10.6	30	12.5	40
6.9	10	9.0	20	10.4	30	12.3	42
6.1	10	8.8	22	11.3	33

1. Create a scatterplot
   • Age is the response variable (y-axis)
   • Diameter is the explanatory variable ( x-axis).  
   • Please make sure your graph is labeled appropriately.
2. Describe the association.
   • Make sure you include, form, direction and strength
3. Create a Best Fit linear model that predicts age from the diameter
   • This needs to be the linear equation
4. Graph this Best Fit linear model on the same graph as the scatterplot.
5. Is it reasonable to predict the age of a tree with this model that is 24 inches in diameter?
6. Find the linear correlation coefficient, r, between the diameter and age.
7. Find the coefficient of determination, R² , and explain what it means in the context of this problem.
   • You need to address the percent variation in the response variable due to the explanatory variable.
8. Create a residual graph
   • Use diameter for the x-axis and  residuals for the y-axis
1. Are the assumptions and conditions met for this linear model to be appropriate for this data?  Please make sure you address the following:
   • Quantitative Variable Condition
   • Straight Enough Condition
   • Outlier Condition
   • Does the Plot Thicken? Condition.

Answer 1

a.

b. Since the correlation coefficient r=0.981897 hence Diameter and age is positively correlated and strength is very strong.

c.

SUMMARY OUTPUT
Regression Statistics
Multiple R	0.981896607
R Square	0.964120946
Adjusted R Square	0.962685784
Standard Error	2.299053955
Observations	27
ANOVA
	df	SS	MS	F	Significance F
Regression	1	3550.821736	3550.822	671.7854	1.38509E-19
Residual	25	132.1412272	5.285649
Total	26	3682.962963
	Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%
Intercept	-16.62383087	1.553112415	-10.7036	7.99E-11	-19.82252576	-13.42513597
Diameter in inches	4.429606435	0.170903072	25.91882	1.39E-19	4.077624968	4.781587901

d.

e. No, since 24 lies outside of the given range of x (i.e. Diameter).

f. The correlation coefficient r=0.981897

g. R²=0.9819 i.e. 98.19% of total variation in Age is explained by this linear regression equation.