The relationship between yield of maize (a type of corn), date of planting, and planting density was investigated in an article. Let the variables be defined as follows. y = maize yield (percent) x1 = planting date (days after April 20) x2 = planting density (10,000 plants/ha) The following regression model with both quadratic terms where x3 = x12 and x4 = x22 provides a good description of the relationship between y and the independent variables. y = ? + ?1 x1 + ?2 x2 + ?3 x3 + ?4 x4 + e (a) If ? = 21.05, ?1 = 0.652, ?2 = 0.0025, ?3 = −0.0204, and ?4 = 0.5, what is the population regression function? y = (b) Use the regression function in part (a) to determine the mean yield (in percent) for a plot planted on May 8 with a density of 41,182 plants/ha. (Round your answer to two decimal places.) % (c) Would the mean yield be higher for a planting date of May 8 or May 22 (for the same density)? The mean yield would be higher for . (d) Is it appropriate to interpret ?1 = 0.652 as the average change in yield when planting date increases by one day and the values of the other three predictors are held fixed? Why or why not? Yes, since there are no other terms involving x1.Yes, since there are other terms involving x1. No, since there are other terms involving x1.No, since there are no other terms involving x1.

MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

Section: Chapter Questions

Problem 1P

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Question

The relationship between yield of maize (a type of corn), date of planting, and planting density was investigated in an article. Let the variables be defined as follows.

y = maize yield (percent)
x₁ = planting date (days after April 20)
x₂ = planting density (10,000 plants/ha)

The following regression model with both quadratic terms where

x₃ = x₁²

and

x₄ = x₂²

provides a good description of the relationship between y and the independent variables.

y = ? + ?₁ x₁ + ?₂ x₂ + ?₃ x₃ + ?₄ x₄ + e

(a)

If ? = 21.05, ?₁ = 0.652, ?₂ = 0.0025,

?₃ = −0.0204,

and

?₄ = 0.5,

what is the population regression function?

y =

(b)

Use the regression function in part (a) to determine the mean yield (in percent) for a plot planted on May 8 with a density of 41,182 plants/ha. (Round your answer to two decimal places.)

(c)

Would the mean yield be higher for a planting date of May 8 or May 22 (for the same density)?

The mean yield would be higher for .

(d)

Is it appropriate to interpret

?₁ = 0.652

as the average change in yield when planting date increases by one day and the values of the other three predictors are held fixed? Why or why not?

Yes, since there are no other terms involving x₁.Yes, since there are other terms involving x₁. No, since there are other terms involving x₁.No, since there are no other terms involving x₁.

Definition Definition Measure of central tendency that is the average of a given data set. The mean value is evaluated as the quotient of the sum of all observations by the sample size. The mean, in contrast to a median, is affected by extreme values. Very large or very small values can distract the mean from the center of the data. Arithmetic mean: The most common type of mean is the arithmetic mean. It is evaluated using the formula: μ = 1 N ∑ i = 1 N x i Other types of means are the geometric mean, logarithmic mean, and harmonic mean. Geometric mean: The nth root of the product of n observations from a data set is defined as the geometric mean of the set: G = x 1 x 2 ... x n n Logarithmic mean: The difference of the natural logarithms of the two numbers, divided by the difference between the numbers is the logarithmic mean of the two numbers. The logarithmic mean is used particularly in heat transfer and mass transfer. ln x 2 − ln x 1 x 2 − x 1 Harmonic mean: The inverse of the arithmetic mean of the inverses of all the numbers in a data set is the harmonic mean of the data. 1 1 x 1 + 1 x 2 + ...

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