578Assignment-5 (Chs. 13 and 14)-solutions:Due by midnight of Sunday, April 26, 2013: drop box 4): 70 points (must show work when possible)
True/False(One point each)
Chapter 13
1. The standard error of the estimate (standard error) is the estimated standard deviation of the distribution of the independent variable (X).
2.In a simple linear regression model, the coefficient of determination only indicates the strength of the relationship between independent and dependent variable, but does not show whether the relationship is positive or negative.
3.When using simple regression analysis, if there is a strong correlation between the independent and dependent variable, then we can conclude that an increase in the value of the independent variable causes an increase in the value of the dependent variable.
4.The error term is the difference between an individual value of the dependent variable and the corresponding mean value of the dependent variable.
5. In bi-variate regression the Coefficient of Determination is always equal to the square of the correlation coefficient. TRUE
6.In Regression Analysis if the variance of the error term is constant, we call it the Heteroscedasticity property.
Chapter 14
7. When the F test is used to test the overall significance of a multiple regression model, if the null hypothesis is rejected, it can be concluded that all of the independent variables X1, X2, Xk are significantly related to the dependent variable Y.
8. An application of the multiple regression model generated the following results involving the F test of the overall regression model: p-value=.0012, R2=.67 and s=.076. Thus, the null hypothesis, which states that none of the independent variables are significantly related to the dependent variable, should be rejected even at the .01 level of significance.
9. High Multicollinearity problem occurs when the Independent variables are highly correlated with the Dependent variable.
10. The assumption of independent error terms in regression analysis is often violated when using time series data and is called the problem of Autocorrelation.
11. Homoscedasticity problem occurs when the assumption of constant error variance is violated.
Multiple Choices(Two points each)
Chapter 13
1. All of the following are assumptions of the error terms in the simple linear regression model except:
A.Errors are normally distributed
B.Error terms have a mean of zero
C.Error terms have a constant variance
D.Error terms depend on the explanatory variable
2.The point estimate of the variance in a regression model is
A.SSE
B.MSE
C.se
D.b1
3. The least squares regression line minimizes the sum of the
A.Sum of Differences between actual and predicted Y values
B.Sum of Squared differences between actual and predicted X values
C.Sum of Absolute deviations between actual and predicted X values
D.Sum of Absolute deviations between actual and predicted Y values
E.Sum of Squared differences between actual and predicted Y values
4.The ______the R2 and the ______the s (standard error), the stronger the relationship between the dependent variable and the independent variable.
A.Higher, lower
B.Lower, higher
C.Lower, lower
D.Higher, higher
5. In simple bivariate regression analysis, if the correlation coefficient is a positive value, then
A.The Y intercept must also be a positive value.
B.The coefficient of determination can be either positive or negative, depending on the value of the slope.
C.The least squares regression equation could either have a positive or a negative slope.
D.The standard error of estimate can either have a positive or a negative value.
E.The slope of the regression line must also be positive.
6. A researcher wants to explore the relationship between the grades students receive on their Midterm test and their Final test score. The following data present the Midterm and Final scores for ten students. What is the correlation coefficient?
Mid / Fin180 / 280
195 / 280
210 / 300
225 / 316
240 / 320
255 / 350
255 / 370
264 / 320
265 / 400
290 / 350
A. 0.556
B. 0.645
C. 0.738
D. 0.802
E. 0.905
Chapter 14
7. Which is not an assumption of a multiple regression model?
A.Positive autocorrelation of error terms
B.Normality of error terms
C.Independence of error terms
D.Constant variation of error terms
E. Independence of error terms with X variables
8. A multiple regression analysis with 22 observations on each of four independent variables and the dependent variable would yield ______and ______degrees of freedom respectively for regression (explained) and error.
A.3, 17
B.4, 20
C.4, 18
D.3, 20
E.4, 17
9. Consider the following partial computer output for a multiple regression model.
What is R2?
A.31.308%
B.76.95%
C.77.72%
D.72.63%
E.23.1%
10. Consider the following partial computer output for a multiple regression model.
What is adjusted R2?
A.31.308%
B.76.95%
C.87.72%
D.72.63%
E.23.1%
11. In multiple regression analysis, the mean square regression divided by mean square error yields the:
A.Standard error
B.F statistic
C.R2
D.Adjusted R2 or
E.T statistic
12. A particular multiple regression model has 3 independent variables, the sum of the squared error is 7680 and the total number of observations is 34. What is the value of the standard error of estimate?
A.256
B.232.72
C.225.89
D.16
E.15.03
Essay Type (Five points each, must show work)
Chapter 13
1. Use the following results obtained from a simple linear regression analysis with 12 observations.
=37.2895-(1.2024)X
r2= 0.6744 and sb1= 0.2934
Test to determine if there is a significant negative relationship between the independent and dependent variable at =.05 and .01
2. A local tire dealer wants to predict the number of tires sold each month. He believes that the number of tires sold is a linear function of the amount of money invested in advertising. He randomly selects 6 months of data consisting of tire sales (in thousands of tires) and advertising expenditures (in thousands of dollars). Based on the data set with 6 observations, the simple linear regression model yielded the following results. (X is advertising expenditure in thousand dollars and Y is tires sold in thousands): ∑X =24; ∑Y =42; ∑X2 = 124; ∑Y2 = 338; ∑XY = 196
Find the Intercept and slope and Write the Regression Equation. Also predict the amount of tires (in thousand tires) sold when money invested in advertising is 5 thousand dollars. Calculate the correlation coefficient, coefficient of determination. Check whether there is a relation between correlation coefficient and coefficient of determination. Calculate SSE and MSE and standard error of the slope coefficient.
3.Consumer Reportsprovided extensive testing and ratings for more than 100 HDTVs. An overall score, based primarily on picture quality, was developed for each model. In general, a higher overall score indicates better performance. The following data show the price and overall score for the ten 42-inch plasma televisions (Consumer Report data slightly changed here):
Brand / Price / ScoreDell / 2800 / 60
Hisense / 2800 / 55
Hitachi / 2700 / 45
JVC / 3500 / 50
LG / 3300 / 55
Maxent / 2000 / 38
Panasonic / 4000 / 67
Phillips / 3000 / 56
Proview / 2500 / 32
Samsung / 3000 / 40
Use the above data to develop an estimated regression equation. Compute the Coefficient of Determination andthe correlation coefficient andshow their relation. Interpret the explanatory power of the model. Estimate the overall score for a 42-inch plasma television with a price of $3400.
Chapter 14
4. A member of the state legislature has expressed concern about the differences in the mathematics test scores of high school freshmen across the state. She asks her research assistant to conduct a study to investigate what factors could account for the differences. The research assistant looked at a random sample of school districts across the state and used the factors of percentage of mathematics teachers in each district with a degree in mathematics, the average age of mathematics teachers and the average salary of mathematics teachers
Analysis of Variance
Write the least squares prediction equation.What is the number of observations in the sample?Based on the multiple regression model given above, estimate the mathematics test score and calculate the value of the residual, if the percentage of teachers with a mathematics degree is 50.0, the average age is 43 and the average salary is 48,300 (48.3). If the actual mathematics test score for these factors is 68.50, what is the error for this observation?
5. For the above equation (question # 4) answer the following: What is the total sum of squares? What is the explained variation?What is the mean square error?
6. For the above equation (question # 4), calculate the Coefficient of Determination and the Adjusted coefficient of Determination and Test for the overall usefulness of the model using F-Statistic at 5% and 1% significance levels.
7. For the above Regression (question # 4), test the usefulness (or significance of the three independent variables using t-test for 5% and 1% significance levels.
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