Consider a regression analysis with three independent variables X₁, X₂, and x3. Select all possible equations for the following regression models. (a) The model that includes as predictors all independent variables but no quadratic or interaction terms. Oy=a + B₁x₁ + B₂×₂ + B3×3 + e Oy = a + B₁x₁ + B₂X₂ + B3X3 + B4x1² + B5×₂² + B6 x 3². + e Oy = a + B₁x₁ + B₂X₂ + B 3×3 + B₁x₁x₂ + e Oy = a + B₁x₁ + B₂X₂ + B3X3 + B4X1 X3 + e С y = a + B₁x₁ + B₂X₂ + B 3×3 + B4X2 X3 + e 2 2 · B₁x₁ + B₂ × ₂ + B 3×3 + B4×₁² + B5×₂² + B6×3² + B7 × ₁ × ₂ + B8 X₁ X3 + B9×₂×3 + e Oy = a + (b) The model that includes as predictors all independent variables and all quadratic terms. Oy = a + B₁x₁ + ₂x₂ + 3x3 + e 2 Oy = a + B₁₂ x₁ + B₂X₂ + B 3×3 + B₁ × ₁² + B 5 x ₂² + B6 x 3² + e Oy = a + B₁x₁ + B₂X₂ + B 3×3 + B₁X1 X₂ + e Oy = a + B₁x₁ + B₂X₂ + B 3×3 + B4X1 X3 + e Oy = a + B₁x₁ + B₂X₂ + B3X3 + B4X₂ X3 + e Oy = a + B₁x₁ + B₂×₂ + B 3×3 + B4×₁² + B 5x₂² + B6 × 3² + B₂×₁×₂ + B8 × 1 ×3 + B9×₂×3 + e (c) All models that include as predictors all independent variables, no quadratic terms, and exactly one interaction term. Oy = a + B₁x₁ + B₂x₂ + 3x3 + e 2 2 Oy = a + B₁x₁ + B₂x₂ + B3X3 + Ba Xí + B5x₂² + B6x3² + e Oy = a + B₁x₁ + B₂X₂ + B3X3 + B4X1X2 + e Oy = a + B₁x₁ + B₂X₂ + B3X3 + B₁X₁ X3 + e Oy = a + B₁x₁ + B₂X₂ + B3x3 + B4×2×3 + e Oy = a + B₁₂ x₁ + B₂X₂ + B3X3 + B4X₁² + B 5 x ₂² + B6 1 (d) The model that includes as predictors all independent variables, all quadratic terms, and all interaction terms (the full quadratic model). Oy=a + B₁x₁ + B₂x₂ + 3x3 + e Oy = a + B₁₂ x₁ + B₂X₂ + B3X3 + B 4x₁² + B 5x₂² Oy = a + B₁x₁ + B₂X₂ + B3X3 + B4x1x₂ + e Oy = a + B₁x₁ + B₂×₂ + B3×3 + B4X₁ X3 + e Oy = a + B₁x₁ + B₂x₂ + B3X3 + B4×2×3 + e Oy = a + B₁₂ x₁ + B₂x₂ + B 3×3 + B4x₁² + B5x₂² + B6X3 +B7x₁x₂ + B8X1 X3 + 9x₂x3 + e +B6x + e + B7 X₁ X₂ + B8X₁ X3 + B9x₂x3 + e

The Simple Linear Regression model is   Y = b0 + b1*X1 + u   and the Multiple Linear Regression model with k variables is:   Y = b0 + b1*X1 + b2*X2 + ... + bk*Xk + u   Y is the dependent variable, the X1, X2, ..., Xk are the explanatory variables, b0 is the intercept, b1, b2, ..., bk are the slope coefficients, and u is the error term,   Yhat represents the OLS fitted values, uhat represent the OLS residuals, b0_hat represents the OLS estimated intercept, and b1_hat, b2_hat,..., bk_hat, represent the OLS estimated slope coefficients. QUESTION 28 Suppose your estimated MLR model is: Y_hat = -30 + 2*X1 + 10*X2 Suppose the standard error for the estimated coefficient associated with X2 is equal to 5. Now, suppose that for some reason we multiply X2 by 5 and we re-estimate the model using the rescaled explanatory variable. What will be the value of the estimated coefficient of X2 and its standard error?   The estimated coefficient of X2 will be equal to 50 and its standard error will be…

A researcher wishes to examine the relationship between years of schooling completed and the number of pregnancies in young women. Her research discovers a linear relationship, and the least squares line is: ŷ = 25x where x is the number of years of schooling completed and y is the number of pregnancies. The slope of the regression line can be interpreted in the following way: When amount of schooling increases by one year, the number of pregnancies tends to increase by 5. When amount of schooling increases by one year, the number of pregnancies tends to decrease by 5. When amount of schooling increases by one year, the number of pregnancies tends to increase by 2. When amount of schooling increases by one year, the number of pregnancies tends to decrease by 2.

The owner of Showtime Movie Theaters, Inc., would like to predict weekly gross revenue as a function of advertising expenditures. Historical data for a sample of eight weeks follow. Weekly Gross Revenue ($1,000s) 96 90 ŷ = 95 92 95 94 94 94 Television Newspaper Advertising Advertising ($1,000s) ($1,000s) 5.0 2.0 4.0 2.5 3.0 3.5 2.5 3.0 1.5 2.0 1.5 2.5 3.3 2.3 4.2 2.5 (a) Develop an estimated regression equation with the amount of television advertising as the independent variable. (Round your numerical values to two decimal places. Let X₁ amount of television advertising in $1,000s and y represent the weekly gross revenue in $1,000s.) ŷ = represent the (b) Develop an estimated regression equation with both television advertising and newspaper advertising as the independent variables. (Round your numerical values to two decimal places. Let x₁ represent the amount of television advertising in $1,000s, x₂ represent the amount of newspaper advertising in $1,000s, and y represent the weekly…

The Simple Linear Regression model is   Y = b0 + b1*X1 + u   and the Multiple Linear Regression model with k variables is:   Y = b0 + b1*X1 + b2*X2 + ... + bk*Xk + u   Y is the dependent variable, the X1, X2, ..., Xk are the explanatory variables, b0 is the intercept, b1, b2, ..., bk are the slope coefficients, and u is the error term,   Yhat represents the OLS fitted values, uhat represent the OLS residuals, b0_hat represents the OLS estimated intercept, and b1_hat, b2_hat,..., bk_hat, represent the OLS estimated slope coefficients. QUESTION 16 In a t-test, suppose a researcher sets the significance level at 0.5%. What does this mean?    The probability that the null hypothesis is true is 0.5% The researcher would be rejecting the null hypothesis, only if the p-value is less than 0.5% The researcher would be rejecting the null hypothesis, if the t-statistic is higher than 0.5 It does not mean anything, because the significance level can only be set at 5%     QUESTION 17 In an MLR…

A study examined the eating habits of 20 children at a nursery school. The variables measured for each child included: calories (the number of calories eaten at lunch), time (the time in minutes spent eating lunch), and sex (male=1, female=0). A multiple linear regression model using Y = calories, X1 = time, and X2 = sex led to the following model: y=547.65-2.85x1+10.67x2   For two children who spend the same amount of time eating, one male and one female, which child is predicted to consume more calories and by how much?

At a large state university, the Statistics department is interested in tracking the progress of its students from entry until graduation. In this example: X represents a student’s final numeric grade (out of 100) in an introductory statistics course Y represents a student’s final numeric grade (out of 100) in an upper-level statistics course   The least-squares regression equation for this relationship is:  Y = 5.20 + 0.93X What is the slope of the regression line? Provide a numeric value as shown in the equation.

A well-known university is interested in how salary (in thousands of dollars) is predicted from years of service for faculty and administrative staff.  Below are the estimated regression equations.Faculty (n = 170): ŷ = 60 + 1.1xAdmin. (n = 155): ŷ = 57 + 1.5x a) How much would a faculty member be earning after 5 years of service? b) In how many years will an administrator earn the same amount as in a)?

The Simple Linear Regression model is       Y = b0 + b1*X1 + u       and the Multiple Linear Regression model with k variables is:       Y = b0 + b1*X1 + b2*X2 + ... + bk*Xk + u       Y is the dependent variable, the X1, X2, ..., Xk are the explanatory variables, b0 is the intercept, b1, b2, ..., bk are the slope coefficients, and u is the error term,       Yhat represents the OLS fitted values, uhat represent the OLS residuals, b0_hat represents the OLS estimated intercept, and b1_hat, b2_hat,..., bk_hat, represent the OLS estimated slope coefficients. QUESTION 7 In the MLR model, the assumption of ‘linearity in parameters’ is violated if:   one of the slope coefficients appears as a power (e.g. Y = b0 + b1*(X1^b2) + b3*X2 + u) the model includes the reciprocal of a variable (e.g. 1/X1) the model includes a variable squared (e.g. X1^2) the model includes a variable in its logarithmic form (i.e. log(X1) )     QUESTION 8 In the MLR model, the assumption of 'no perfect collinearity'…