App. Stat Test 2

Exhibit 13-8 The following estimated regression model was developed relating yearly income (Y in $1,000s) of 30 individuals with their age (X1) and their gender (X2) (0 if male and 1 if female).Also provided are SST = 1,200 and SSE = 384. Refer to Exhibit 13-8. The yearly income of a 24-year-old male individual is

$46,800

Exhibit 13-8 The following estimated regression model was developed relating yearly income (Y in $1,000s) of 30 individuals with their age (X1) and their gender (X2) (0 if male and 1 if female).Also provided are SST = 1,200 and SSE = 384. Refer to Exhibit 13-8. The estimated income of a 30-year-old male is

$51,000

Regression analysis was applied between sales (Y in $1,000) and advertising (X in $100), and the following estimated regression equation was obtained. = 80 + 6.2 XBased on the above estimated regression line, if advertising is $10,000, then the point estimate for sales (in dollars) is

$700,00

Regression analysis was applied between sales (in $1000) and advertising (in $100) and the following regression function was obtained. = 500 + 4 XBased on the above estimated regression line if advertising is $10,000, then the point estimate for sales (in dollars) is

$900,000

Exhibit 12-7You are given the following information about y and x. yDependent Variable xIndependent Variable 5 4 7 6 9 2 11 4 Refer to Exhibit 12-7. The least squares estimate of b1 (slope) equals

-0.5

Exhibit 12-2 You are given the following information about y and x. Dependent Variable (Y) Independent Variable (X) 5 1 4 2 3 3 2 4 1 5 Refer to Exhibit 12-2. The sample correlation coefficient equals

-1

Exhibit 13-6Below you are given a partial computer output based on a sample of 16 observations. Coefficient Standard Error Constant 12.924 4.425 X1 -3.682 2.630 X2 45.216 12.560 Analysis of Variance Source ofVariation Degreesof Freedom Sum ofSquares MeanSquare F Regression 4,853 2,426.5 Error 485.3 Refer to Exhibit 13-6. We want to test whether the parameter β1 is significant. The test statistic equals

-1.4

Exhibit 13-11Below you are given a partial computer output based on a sample of 25 observations. Coefficient Standard Error Constant 145 29 X1 20 5 X2 -18 6 X3 4 4 Refer to Exhibit 13-11. We want to test whether the parameter β2 is significant. The test statistic equals

-3

Exhibit 12-10The following information regarding a dependent variable Y and an independent variable X is provided. n = 4 ΣX = 16 ΣY = 28 Σ (Y -)(X - ) = -8 Σ (X - )2 = 8 SST = 42 SSE = 34 Refer to Exhibit 12-10. The coefficient of determination is

0.1905

Exhibit 13-12 In a laboratory experiment, data were gathered on the life span (Y in months) of 33 rats, units of daily protein intake (X1), and whether or not agent X2 (a proposed life extending agent) was added to the rats diet (X2 = 0 if agent X2 was not added, and X2 = 1 if agent was added.) From the results of the experiment, the following regression model was developed.Also provided are SSR = 60 and SST = 180. Refer to Exhibit 13-12. The multiple coefficient of determination is

0.333

Exhibit 13-1In a regression model involving 44 observations, the following estimated regression equation was obtained.For this model SSR = 600 and SSE = 400. Refer to Exhibit 13-1. The coefficient of determination for the above model is

0.600

Exhibit 12-1The following information regarding a dependent variable (Y) and an independent variable (X) is provided. Y X 4 2 3 1 4 4 6 3 8 5 SSE = 6SST = 16 Refer to Exhibit 12-1. The coefficient of determination is

0.625

Exhibit 12-9A regression and correlation analysis resulted in the following information regarding a dependent variable (y) and an independent variable (x). n = 10 ΣX = 90 ΣY = 170 Σ (Y -)(X - ) = = 466 Σ (Y - )2 = = 1434 Σ (X - )2 = 234 SSE = 505.98 Refer to Exhibit 12-9. The coefficient of determination equals

0.6472

Exhibit 13-8 The following estimated regression model was developed relating yearly income (Y in $1,000s) of 30 individuals with their age (X1) and their gender (X2) (0 if male and 1 if female).Also provided are SST = 1,200 and SSE = 384. Refer to Exhibit 13-8. The multiple coefficient of determination is

0.68

In a regression analysis involving 25 observations, the following estimated regression equation was developed.Also, the following standard errors and the sum of squares were obtained. Sb1 = 3Sb2 = 6Sb3 = 7SST = 4,800SSE = 1,296 Refer to Exhibit 13-9. The multiple coefficient of determination is

0.73

Exhibit 13-7A regression model involving 4 independent variables and a sample of 15 periods resulted in the following sum of squares. SSR = 165 SSE = 60 Refer to Exhibit 13-7. The coefficient of determination is

0.7333

For a multiple regression model, SSR = 600 and SSE = 200. The multiple coefficient of determination is

0.75

For a multiple regression model, SST = 200 and SSE = 50. The multiple coefficient of determination is

0.75

Exhibit 13-10In a regression model involving 30 observations, the following estimated regression equation was obtained.For this model, SSR = 1,740 and SST = 2,000. Refer to Exhibit 13-10. The coefficient of determination for this model is

0.8700

Exhibit 13-3In a regression model involving 30 observations, the following estimated regression equation was obtained:For this model SSR = 700 and SSE = 100. Refer to Exhibit 13-3. The coefficient of determination for the above model is approximately

0.875

In a multiple regression analysis involving 12 independent variables and 166 observations, SSR = 878 and SSE = 122. The coefficient of determination is

0.878

In a multiple regression analysis involving 5 independent variables and 30 observations, SSR = 360 and SSE = 40. The coefficient of determination is

0.90

Exhibit 12-1The following information regarding a dependent variable (Y) and an independent variable (X) is provided. Y X 4 2 3 1 4 4 6 3 8 5 SSE = 6SST = 16 Refer to Exhibit 12-1. The least squares estimate of the slope is

1

Exhibit 13-10 In a regression model involving 30 observations, the following estimated regression equation was obtained.For this model, SSR = 1,740 and SST = 2,000. Refer to Exhibit 13-10. The value of MSE is

10.83

Exhibit 12-10The following information regarding a dependent variable Y and an independent variable X is provided. n = 4 ΣX = 16 ΣY = 28 Σ (Y -)(X - ) = -8 Σ (X - )2 = 8 SST = 42 SSE = 34 Refer to Exhibit 12-10. The Y intercept is

11

Exhibit 12-3You are given the following information about y and x. yDependent Variable xIndependent Variable 12 4 3 6 7 2 6 4 Refer to Exhibit 12-3. The least squares estimate of b0 equals

11

In order to test for the significance of a regression model involving 14 independent variables and 255 observations, the numerator and denominator degrees of freedom (respectively) for the critical value of F are

14 and 240

Exhibit 12-10The following information regarding a dependent variable Y and an independent variable X is provided. n = 4 ΣX = 16 ΣY = 28 Σ (Y -)(X - ) = -8 Σ (X - )2 = 8 SST = 42 SSE = 34 Refer to Exhibit 12-10. The MSE is

17

Exhibit 13-2 A regression model between sales (Y in $1,000), unit price (X1 in dollars) and television advertisement (X2 in dollars) resulted in the following function:For this model SSR = 3500, SSE = 1500, and the sample size is 18. Refer to Exhibit 13-2. To test for the significance of the model, the test statistic F is

17.5

Exhibit 13-9 In a regression analysis involving 25 observations, the following estimated regression equation was developed.Also, the following standard errors and the sum of squares were obtained. Sb1 = 3 Sb2 = 6 Sb3 = 7 SST = 4,800 SSE = 1,296 Refer to Exhibit 13-9. The test statistic for testing the significance of the model is

18.926

regression model involved 18 independent variables and 200 observations. The critical value of t for testing the significance of each of the independent variable's coefficients will have

181 degrees of freedom

Exhibit 12-1The following information regarding a dependent variable (Y) and an independent variable (X) is provided. Y. X 4. 2 3 1 4 4 6 3 8 5 SSE = 6SST = 16 Refer to Exhibit 12-1. The MSE is

2

Exhibit 13-12 In a laboratory experiment, data were gathered on the life span (Y in months) of 33 rats, units of daily protein intake (X1), and whether or not agent X2 (a proposed life extending agent) was added to the rats diet (X2 = 0 if agent X2 was not added, and X2 = 1 if agent was added.) From the results of the experiment, the following regression model was developed.Also provided are SSR = 60 and SST = 180. Refer to Exhibit 13-12. The degrees of freedom associated with SSR are

2

Exhibit 13-5 Below you are given a partial Minitab output based on a sample of 25 observations. Coefficient Standard Error Constant 145.321 48.682 X1 25.625 9.150 X2 -5.720 3.575 X3 0.823 0.183 Refer to Exhibit 13-5. The t value obtained from the table to test an individual parameter at the 5% level is

2.080

Exhibit 13-9In a regression analysis involving 25 observations, the following estimated regression equation was developed.Also, the following standard errors and the sum of squares were obtained. Sb1 = 3 Sb2 = 6 Sb3 = 7 SST = 4,800 SSE = 1,296 Refer to Exhibit 13-9. If you want to determine whether or not the coefficients of the independent variables are significant, the critical value of t statistic at α = 0.05 is

2.080

Exhibit 13-1 In a regression model involving 44 observations, the following estimated regression equation was obtained.For this model SSR = 600 and SSE = 400. Refer to Exhibit 13-1. The computed F statistics for testing the significance of the above model is

20.00

Exhibit 13-10 In a regression model involving 30 observations, the following estimated regression equation was obtained.For this model, SSR = 1,740 and SST = 2,000. Refer to Exhibit 13-10. The value of SSE is

260

Exhibit 13-8 The following estimated regression model was developed relating yearly income (Y in $1,000s) of 30 individuals with their age (X1) and their gender (X2) (0 if male and 1 if female).Also provided are SST = 1,200 and SSE = 384. Refer to Exhibit 13-8. The test statistic for testing the significance of the model is

28.69

Exhibit 13-6Below you are given a partial computer output based on a sample of 16 observations. Coefficient Standard Error Constant 12.924 4.425 X1 -3.682 2.630 X2 45.216 12.560 Analysis of Variance Source ofVariation Degreesof Freedom Sum ofSquares MeanSquare F Regression 4,853 2,426.5 Error 485.3 Refer to Exhibit 13-6. The t value obtained from the table which is used to test an individual parameter at the 1% level is

3.012

Exhibit 13-9 In a regression analysis involving 25 observations, the following estimated regression equation was developed.Also, the following standard errors and the sum of squares were obtained. Sb1 = 3Sb2 = 6Sb3 = 7SST = 4,800SSE = 1,296 Refer to Exhibit 13-9. If we are interested in testing for the significance of the relationship among the variables (i.e., significance of the model) the critical value of F at α = 0.05 is

3.07

Exhibit 13-12In a laboratory experiment, data were gathered on the life span (Y in months) of 33 rats, units of daily protein intake (X1), and whether or not agent X2 (a proposed life extending agent) was added to the rats diet (X2 = 0 if agent X2 was not added, and X2 = 1 if agent was added.) From the results of the experiment, the following regression model was developed.Also provided are SSR = 60 and SST = 180. Refer to Exhibit 13-12. If we want to test for the significance of the model, the critical value of F at 95% confidence is

3.32

Exhibit 13-8The following estimated regression model was developed relating yearly income (Y in $1,000s) of 30 individuals with their age (X1) and their gender (X2) (0 if male and 1 if female).Also provided are SST = 1,200 and SSE = 384. Refer to Exhibit 13-8. If we want to test for the significance of the model, the critical value of F at 95% confidence is

3.35

Exhibit 13-7A regression model involving 4 independent variables and a sample of 15 periods resulted in the following sum of squares. SSR = 165 SSE = 60 Refer to Exhibit 13-7. If we want to test for the significance of the model at 95% confidence, the critical F value (from the table) is

3.48

Exhibit 13-6 Below you are given a partial computer output based on a sample of 16 observations. Coefficient Standard Error Constant 12.924 4.425 X1 -3.682 2.630 X2 45.216 12.560 Analysis of Variance Source ofVariation Degreesof Freedom Sum ofSquares MeanSquare F Regression 4,853 2,426.5 Error 485.3 Refer to Exhibit 13-6. The F value obtained from the table used to test if there is a relationship among the variables at the 5% level equals

3.81

Exhibit 13-12 In a laboratory experiment, data were gathered on the life span (Y in months) of 33 rats, units of daily protein intake (X1), and whether or not agent X2 (a proposed life extending agent) was added to the rats diet (X2 = 0 if agent X2 was not added, and X2 = 1 if agent was added.) From the results of the experiment, the following regression model was developed.Also provided are SSR = 60 and SST = 180.

30

Exhibit 13-10 In a regression model involving 30 observations, the following estimated regression equation was obtained.For this model, SSR = 1,740 and SST = 2,000. Refer to Exhibit 13-10. The test statistic F for testing the significance of the above model is

32.12

Exhibit 13-12In a laboratory experiment, data were gathered on the life span (Y in months) of 33 rats, units of daily protein intake (X1), and whether or not agent X2 (a proposed life extending agent) was added to the rats diet (X2 = 0 if agent X2 was not added, and X2 = 1 if agent was added.) From the results of the experiment, the following regression model was developed.Also provided are SSR = 60 and SST = 180. Refer to Exhibit 13-12. The life expectancy of a rat that was given 2 units of agent X2 daily, but was not given any protein is

34.3

Exhibit 13-10In a regression model involving 30 observations, the following estimated regression equation was obtained.For this model, SSR = 1,740 and SST = 2,000. Refer to Exhibit 13-10. The value of MSR is

348

In order to test for the significance of a regression model involving 4 independent variables and 36 observations, the numerator and denominator degrees of freedom (respectively) for the critical value of F are

4 and 31

Exhibit 13-10 In a regression model involving 30 observations, the following estimated regression equation was obtained.For this model, SSR = 1,740 and SST = 2,000. Refer to Exhibit 13-10. The degrees of freedom associated with SSR are

5

Exhibit 12-2You are given the following information about y and x. Dependent Variable (Y) Independent Variable (X) 5 1 4 2 3 3 2 4 1 5 Refer to Exhibit 12-2. The least squares estimate of b0 (intercept)equals

6

Exhibit 12-10The following information regarding a dependent variable Y and an independent variable X is provided. n = 4 ΣX = 16 ΣY = 28 Σ (Y -)(X - ) = -8 Σ (X - )2 = 8 SST = 42 SSE = 34 Refer to Exhibit 12-10. The point estimate of Y when X = 3 is

8

In order to test for the significance of a regression model involving 8 independent variables and 121 observations, the numerator and denominator degrees of freedom (respectively) for the critical value of F are

8 and 112

Exhibit 13-4 a. b. c. d. Refer to Exhibit 13-4. Which equation gives the estimated regression line?

Equation C

Exhibit 13-4a. b. c. d. Refer to Exhibit 13-4. Which equation describes the multiple regression equation?

Equation D

All the variables in a multiple regression analysis

None of these alternatives is correct.

Exhibit 13-10 In a regression model involving 30 observations, the following estimated regression equation was obtained.For this model, SSR = 1,740 and SST = 2,000. Refer to Exhibit 13-10. The degrees of freedom associated with SST are

None of these alternatives is correct.

If there is a very weak correlation between two variables, then the coefficient of determination must be

None of these alternatives is correct.

The mathematical equation relating the expected value of the dependent variable to the value of the independent variables, which has the form of E(y) = is

a multiple regression equation

A variable that cannot be measured in numerical terms is called

a qualitative variable

The estimate of the multiple regression equation based on the sample data, which has the form of E(y) = is

an estimated multiple regression equation

In a multiple regression model, the error term ε is assumed to

be normally distributed

Exhibit 13-6Below you are given a partial computer output based on a sample of 16 observations. Coefficient Standard Error Constant 12.924 4.425 X1 -3.682 2.630 X2 45.216 12.560 Analysis of Variance Source ofVariation Degreesof Freedom Sum ofSquares MeanSquare F Regression 4,853 2,426.5 Error 485.3 Refer to Exhibit 13-6. Carry out the test to determine if there is a relationship among the variables at the 5% level. The null hypothesis should

be rejected

Exhibit 12-4 Regression analysis was applied between sales data (Y in $1,000s) and advertising data (x in $100s) and the following information was obtained. = 12 + 1.8 xn = 17SSR = 225SSE = 75Sb1 = 0.2683 Refer to Exhibit 12-4. To perform an F test, the p-value is

between .05 and 0.1

Exhibit 13-12 In a laboratory experiment, data were gathered on the life span (Y in months) of 33 rats, units of daily protein intake (X1), and whether or not agent X2 (a proposed life extending agent) was added to the rats diet (X2 = 0 if agent X2 was not added, and X2 = 1 if agent was added.) From the results of the experiment, the following regression model was developed.Also provided are SSR = 60 and SST = 180. Refer to Exhibit 13-12. The p-value for testing the significance of the regression model is

between 0.01 and 0.025

If the coefficient of determination is equal to 1, then the coefficient of correlation

can be either -1 or +1

If the coefficient of determination is a positive value, then the coefficient of correlation

can be either negative or positive

The value of the coefficient of correlation (R)

can be equal to the value of the coefficient of determination (R2)

The numerical value of the coefficient of determination

can be larger or smaller than the coefficient of correlation

In regression and correlation analysis, if SSE and SST are known, then with this information the

coefficient of determination can be computed

A multiple regression model has the form Y = 12 - 8X1 + 3X2 As X1 increases by 2 units (holding X2 constant), Y is expected to

decrease by 16 units

Regression analysis was applied between demand for a product (Y) and the price of the product (X), and the following estimated regression equation was obtained. = 120 - 10 XBased on the above estimated regression equation, if price is increased by 2 units, then demand is expected to

decrease by 20 units

In regression analysis, the response variable is the

dependent variable

In regression analysis, the variable that is being predicted is the

dependent variable

The model developed from sample data that has the form of is known as

estimated regression equation

Exhibit 13-8 The following estimated regression model was developed relating yearly income (Y in $1,000s) of 30 individuals with their age (X1) and their gender (X2) (0 if male and 1 if female).Also provided are SST = 1,200 and SSE = 384. Refer to Exhibit 13-8. From the above function, it can be said that the expected yearly income of

females is $3,000 more than males

A model in the form of y = β0 + β1z1 + β2z2 + . . . +βpzp + ε where each independent variable zj (for j = 1, 2, . . ., p) is a function of xj . xj is known as the

general linear model

A multiple regression model has the formY = 70 - 14X1 + 5X2 As X1decreases by 1 unit (holding X2 constant), Y is expected to

increase by 14 units

A multiple regression model has the formY = 7 + 2X1 + 9X2 As X1 increases by 1 unit (holding X2 constant), Y is expected to

increase by 2 units

A regression analysis between sales (Y in $1000) and advertising (X in dollars) resulted in the following equation = 30,000 + 4 XThe above equation implies that an

increase of $1 in advertising is associated with an increase of $4,000 in sales

If the coefficient of correlation is 0.4, the percentage of variation in the dependent variable explained by the variation in the independent variable

is 16%

Exhibit 13-9 In a regression analysis involving 25 observations, the following estimated regression equation was developed.Also, the following standard errors and the sum of squares were obtained. Sb1 = 3Sb2 = 6Sb3 = 7SST = 4,800SSE = 1,296 Refer to Exhibit 13-9. The coefficient of X2

is not significant

Exhibit 13-9 In a regression analysis involving 25 observations, the following estimated regression equation was developed.Also, the following standard errors and the sum of squares were obtained. Sb1 = 3Sb2 = 6Sb3 = 7SST = 4,800SSE = 1,296 Refer to Exhibit 13-9. The coefficient of X3

is not significant

Exhibit 13-9 In a regression analysis involving 25 observations, the following estimated regression equation was developed.Also, the following standard errors and the sum of squares were obtained. Sb1 = 3Sb2 = 6Sb3 = 7SST = 4,800SSE = 1,296 Refer to Exhibit 13-9. The coefficient of X1

is significant

The coefficient of correlation

is the square root of the coefficient of determination

SSE can never be

larger than SST

Exhibit 13-10 In a regression model involving 30 observations, the following estimated regression equation was obtained.For this model, SSR = 1,740 and SST = 2,000. Refer to Exhibit 13-10. The p-value for testing the significance of the regression model is

less than 0.01

Exhibit 13-9In a regression analysis involving 25 observations, the following estimated regression equation was developed.Also, the following standard errors and the sum of squares were obtained. Sb1 = 3Sb2 = 6Sb3 = 7SST = 4,800SSE = 1,296 Refer to Exhibit 13-9. The p-value for testing the significance of the regression model is

less than 0.01

Exhibit 13-3In a regression model involving 30 observations, the following estimated regression equation was obtained:For this model SSR = 700 and SSE = 100. Refer to Exhibit 13-3. The conclusion is that the

model is significant

A multiple regression model has

more than one independent variable

A term used to describe the case when the independent variables in a multiple regression model are correlated is

multicollinearity

A measure of goodness of fit for the estimated regression equation is the

multiple coefficient of determination

If the coefficient of correlation is a positive value, then the regression equation

must have a positive slope.

Exhibit 13-6Below you are given a partial computer output based on a sample of 16 observations. Coefficient Standard Error Constant 12.924 4.425 X1 -3.682 2.630 X2 45.216 12.560 Analysis of Variance Source ofVariation Degreesof Freedom Sum ofSquares MeanSquare F Regression 4,853 2,426.5 Error 485.3 Refer to Exhibit 13-6. Carry out the test of significance for the parameter β1 at the 1% level. The null hypothesis should be

not rejected

The following modelis referred to as a

simple first-order model with one predictor variable

The standard error is the

square root of MSE

If only MSE is known, you can compute the

standard error

In a multiple regression model, the variance of the error term ε is assumed to be

the same for all values of the independent variable

In a multiple regression model, the error term ε is assumed to be a random variable with a mean of

zero

In multiple regression analysis, the word linear in the term "general linear model" refers to the fact that

β0, β1, . . . βp, all have exponents of 1