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Chapter 12: Q5E (page 722)

Question:How is the number of degrees of freedom available for estimating 蟽2(the variance of蔚 ) related to the number of independent variables in a regression model?

Short Answer

Expert verified

To estimate the value of the coefficients of any K+1 independent variable,K+1 numbers of equations are needed to mathematically solve it and find unique solutions to the equations

Step by step solution

01

Step-by-Step SolutionStep 1: No of independent variables in the model

To estimate the value of the coefficients of any K+1 independent variable,K+1numbers of equations are needed to mathematically solve it and find unique solutions to the equations

02

Degrees of freedom

Therefore, the no of degrees of freedom available for estimating the variance of 蟽2, would be n-(k+1).

Degree of freedom: degree of freedom is the number of values which are allowed to vary in the final calculation of a statistic.

In this case, the variance of error term is under discussion.

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Most popular questions from this chapter

Assertiveness and leadership. Management professors at Columbia University examined the relationship between assertiveness and leadership (Journal of Personality and Social Psychology, February 2007). The sample represented 388 people enrolled in a full-time MBA program. Based on answers to a questionnaire, the researchers measured two variables for each subject: assertiveness score (x) and leadership ability score (y). A quadratic regression model was fit to the data, with the following results:

a. Conduct a test of overall model utility. Use伪=0.05 .

b. The researchers hypothesized that leadership ability increases at a decreasing rate with assertiveness. Set up the null and alternative hypotheses to test this theory.

  1. Use the reported results to conduct the test, part b. Give your conclusion(at伪=0.05 )in the words of the problem.

Question: Predicting elements in aluminum alloys. Aluminum scraps that are recycled into alloys are classified into three categories: soft-drink cans, pots and pans, and automobile crank chambers. A study of how these three materials affect the metal elements present in aluminum alloys was published in Advances in Applied Physics (Vol. 1, 2013). Data on 126 production runs at an aluminum plant were used to model the percentage (y) of various elements (e.g., silver, boron, iron) that make up the aluminum alloy. Three independent variables were used in the model: x1 = proportion of aluminum scraps from cans, x2 = proportion of aluminum scraps from pots/pans, and x3 = proportion of aluminum scraps from crank chambers. The first-order model, , was fit to the data for several elements. The estimates of the model parameters (p-values in parentheses) for silver and iron are shown in the accompanying table.

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(b)Is the overall model statistically useful (at a = .05) for predicting the percentage of iron in the alloy? If so, give a practical interpretation of R2.

(c)Based on the parameter estimates, sketch the relationship between percentage of silver (y) and proportion of aluminum scraps from cans (x1). Conduct a test to determine if this relationship is statistically significant at 伪 = .05.

(d)Based on the parameter estimates, sketch the relationship between percentage of iron (y) and proportion of aluminum scraps from cans (x1). Conduct a test to determine if this relationship is statistically significant at 伪 = .05.

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E(y)=尾0+尾1x1+尾2x12+尾3x2+尾4x1x2+尾5x12x2herex2=1E(y)=尾0+尾1x1+尾2x12+尾3(1)+尾4x1(1)+尾5(1)2(1)E(y)=(尾0+尾3)+(尾1+尾4)x1+(尾2+尾5)(x1)2

a. For jobs that are not highly complex, write the equation of the model for E1y2 as a function of x1. (Substitute x2 = 0 into the equation.)

b. Refer to part a. What do each of the b鈥檚 represent in the model?

c. For highly complex jobs, write the equation of the model for E(y) as a function of x1. (Substitute x2 = 1 into the equation.)

d. Refer to part c. What do each of the b鈥檚 represent in the model?

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a) Give the equation of a first-order model relating revenue (y)to both tweet rate(x1)and PN-ratio(x2).

b) Which b in the model, part a, represents the change in revenue(y)for every 1-tweet increase in the tweet rate(x1), holding PN-ratio(x2)constant?

c) Which b in the model, part a, represents the change in revenue (y)for every 1-unit increase in the PN-ratio(x2), holding tweet rate(x1)constant?

d) The following coefficients were reported:R2=0.945andRa2=0.940. Give a practical interpretation for bothR2andRa2.

e) Conduct a test of the null hypothesis, H0;尾1=尾2=0. Use伪=0.05.

f) The researchers reported the p-values for testing,H0;尾1=0andH0;尾2=0 as both less than .0001. Interpret these results (use).

Suppose you fit the quadratic model E(y)=尾0+尾1x+尾2x2to a set of n = 20 data points and found R2=0.91, SSyy=29.94, and SSE = 2.63.

a. Is there sufficient evidence to indicate that the model contributes information for predicting y? Test using a = .05.

b. What null and alternative hypotheses would you test to determine whether upward curvature exists?

c. What null and alternative hypotheses would you test to determine whether downward curvature exists?
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