Measuring tree height is not an easy task. How well might trunk diameter predict

Question

Measuring tree height is not an easy task. How well might trunk diameter predict tree height? A survey of 958 live trees in an old-growth forest in Canada answered this question. Here is part of the computer output based on these data for regressing height on diameter, both in centimeters (cm), along with prediction for a tree having a diameter of 50 cm:

(a) A scatterplot of the data shows a reasonably linear relationship between tree height and diameter. Is this relationship statistically significant? How strong is the relationship?

(b) Give a 95% confidence interval for the height of 1 tree randomly selected from this forest if the tree has a diameter of 50 cm.

(c) Now give a 95% confidence interval for the mean height of all the trees in this forest that have a diameter of 50 cm. How does this interval compare with the one you calculated in (b)?

please show your work and do not use a statistical software.

Predictor Coef SE Coef Constant 2.6696 0.1677 15.92 0.000 Diameter 0.550940 0.005058 108.93 0.000 S 3.67427 R-Sq 92.5% R-Sq (adj) 92.5% New 95% CI 95% PI Obs Fit SE Fit 1 30.217 0.179 (29.865.30.569) (22.997.37.436)

Explanation / Answer

Part-a

As coefficient of Diameter is significant with p-value=0.000<0.05, so the relationship is statistically significant. The relationship is very strong as diameter explain (R-square) 92.5% of the variability in tree height.

Part-b

Predicted height yhat= 2.6696+0.550940*50=30.2166

From given output we have

95% prediction interval =(22.997   , 37.436)

Part-c

From given output

95% confidence Interval of mean=(29.865 , 30.569)

Confidence interval is narrower than prediction interval as predicting mean has smaller standard error than predicting single value.

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