A logging company wants to calculate the total amount of volume in cubic meters of wood, in logs that can be extracted from a eucalyptus forest. The perimeter of the log is easy to obtain, but the height is difficult to measure. Not even with the use of an appropriate instrument is it possible to measure because the trees are too close to each other. The forestry engineer in charge decides to try a sampling estimate. The idea is to measure the trunk perimeter and height (until the beginning of the branch) of 13 randomly selected trees and extend the result obtained to all 2000 trees in the forest. It is known that there is a constant relationship between the height (Y) and the perimeter of the trunk (X) of the tree. Thus, the engineer proposes that 13 trees with trunks of different perimeters be selected and their heights measured. Next, he proposes to fit a linear regression model to the data obtained yi = a + bxi + ei. The model will provide a value that establishes the relationship between the height and perimeter of the trunk. In this way, once the trunk's perimeter is measured, an estimated value for the height of each of the 2000 trees in the forest will be calculated. With the perimeter and height values, we can have a very accurate estimate of the total volume of wood available. For the sample of 13 trees, the values in the image below can be seen: Tip: Round the final answer value to 4 decimal places, but do not round off in the middle stages of resolution. a) Calculate the sample correlation coefficient between x and y: b) Consider the simple regression model yi = α + βxi + εi to model the relationship between the variables tree height (Y) in cm and perimeter (X) in cm. Based on the sample, calculate an estimate for the coefficient β of the line. c) Use the regression line yi = α + βxi to estimate the height of a tree measuring at the trunk perimeter 54 cm. d) After the 2000 trees have been cut, the company is able to deal immediately with only 500 of them (25%), being necessary to store the rest. The company chooses to treat the longest trees soon. She has to build a shed to store the 75% shortest trees. Based on the sample, what would be the estimated height of the largest tree to be stored in the shed?
A logging company wants to calculate the total amount of volume in cubic meters of wood, in logs that can be extracted from a eucalyptus forest. The perimeter of the log is easy to obtain, but the height is difficult to measure. Not even with the use of an appropriate instrument is it possible to measure because the trees are too close to each other. The forestry engineer in charge decides to try a sampling estimate. The idea is to measure the trunk perimeter and height (until the beginning of the branch) of 13 randomly selected trees and extend the result obtained to all 2000 trees in the forest. It is known that there is a constant relationship between the height (Y) and the perimeter of the trunk (X) of the tree. Thus, the engineer proposes that 13 trees with trunks of different perimeters be selected and their heights measured. Next, he proposes to fit a linear regression model to the data obtained yi = a + bxi + ei. The model will provide a value that establishes the relationship between the height and perimeter of the trunk. In this way, once the trunk's perimeter is measured, an estimated value for the height of each of the 2000 trees in the forest will be calculated. With the perimeter and height values, we can have a very accurate estimate of the total volume of wood available. For the sample of 13 trees, the values in the image below can be seen:
Tip: Round the final answer value to 4 decimal places, but do not round off in the middle stages of resolution.
a) Calculate the sample
b) Consider the simple regression model yi = α + βxi + εi to model the relationship between the variables tree height (Y) in cm and perimeter (X) in cm. Based on the sample, calculate an estimate for the coefficient β of the line.
c) Use the regression line yi = α + βxi to estimate the height of a tree measuring at the trunk perimeter 54 cm.
d) After the 2000 trees have been cut, the company is able to deal immediately with only 500 of them (25%), being necessary to store the rest. The company chooses to treat the longest trees soon. She has to build a shed to store the 75% shortest trees. Based on the sample, what would be the estimated height of the largest tree to be stored in the shed?
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