Stat 401, Section F Homework 9
Due Date: Wednesday, October 31
1. The rowan (Sorbus aucuparia) is a tree that grows in a wide range of altitudes. To study how the tree
adapts to its varying habitats, researchers collected twigs with attached buds from 12 trees growing at
various altitudes in North Angus, Scotland. The buds were brought back to the laboratory and measure-
ments were made on the dark respiration rate. The average altitude from the origin for the 12 trees is
433.33 meters. The average respiration rate computed for the 12 trees is 0.21 µl oxygen per hour per
mg dry weight of tissue. The standard deviation of the altitude for the 12 trees is 214.62 meters and for
respiration rate is 0.077 µl/hr-mg. The correlation between altitude from the origin and dark respiration
rate was 0.887.
(a) We will use linear regression to describe the relationship between altitude and dark respiration.
Which of the two variables would most naturally be considered the explanatory variable and which
would be the response variable?
(b) Compute the equation of the least-squares regression line.
(c) Compute a 95% confidence interval for the slope of the regression line.
(d) Estimate the mean respiration rate for a tree growing at 300 meters above the origin.
(e) Do you think your estimate in part (d) involved interpolation or extrapolation? Explain.
(f) Estimate the standard deviation of mean dark respiration rate for trees growing 300 meters above
the origin.
2. It is generally thought that the percentage of fruits attacked by codling moth larvae is greater on apple
trees bearing a small crop. Apparently the density of the flying moths is unrelated to the size of the crop
on a tree, so the chance of attack for any particular fruit is increased if few fruits are on the tree. Data
on a sample of 10 trees gives a sample linear correlation coefficient of -0.8. Other summary statistics
obtained from the sample of 10 trees are provided below.
Variable Sample Mean Sample Standard Deviation
Crop Size (number of fruit) 110 40
Percentage of Wormy Fruits 45 12
(a) Find the equation of the least-squares regression line for predicting percentage of wormy fruits from
crop size.
(b) Write down the ANOVA table for the simple linear regression of percentage of wormy fruit on crop
size. Compute the Fstatistic and find a p-value.
(c) Based on the Fstatistic and p-value that you computed in part (b), would you say the slope of the
regression line is significantly different from 0?
(d) Find a 99% confidence interval for β1, the slope of the least squares regression line for predicting
percentage of wormy fruit from crop size.
(e) What proportion of the variability in percentage of wormy fruit is explained by the regression of
percentage of wormy fruit on crop size?
(f) Estimate the mean percentage of wormy fruits for trees with a crop size of 150 fruit.
(g) Provide a 95% confidence interval for the mean percentage of wormy fruits for trees with a crop
size of 150 fruit.
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