Question

The owner of a moving company typically has his most experienced manager predict the total number of labor hours that will be required to complete an upcoming move. This approach has proved useful in the past, but the owner has the business objective of developing a more accurate method of predicting labor hours. In a preliminary effort to provide a more accurate method, the owner has decided to use the number of cubic feet moved and whether there is an elevator in the apartment building as the independent variables and has collected data for 36 moves in which the origin and destination were within the borough of Manhattan in New York City and the travel time was an insignificant portion of the hours worked. The data are organized and stored in. For (a) through (k), do not include an interaction term Moving. a. State the multiple regression equation for predicting labor hours, using the number of cubic feet moved and whether there is an elevator. b. Interpret the regression coefficients in (a). c. Predict the labor hours for moving 500 cubic feet in an apartment building that has an elevator and construct a $95 \%$ confidence interval estimate and a $95 \%$ prediction interval. d. Perform a residual analysis on the results and determine whether the regression assumptions are valid. e. Is there a significant relationship between labor hours and the two independent variables (cubic feet moved and whether there is an elevator in the apartment building) at the 0.05 level of significance? f. At the 0.05 level of significance, determine whether each independent variable makes a contribution to the regression model. Indicate the most appropriate regression model for this set of data. g. Construct a $95 \%$ confidence interval estimate of the population slope for the relationship between labor hours and cubic feet moved. h. Construct a $95 \%$ confidence interval estimate for the relationship between labor hours and the presence of an elevator. i. Compute and interpret the adjusted $r^2$. j. Compute the coefficients of partial determination and interpret their meaning. k. What assumption do you need to make about the slope of labor hours with cubic feet moved? 1. Add an interaction term to the model and, at the 0.05 level of significance, determine whether it makes a significant contribution to the model. m. On the basis of the results of (f) and (1), which model is most appropriate? Explain.

The owner of a moving company typically has his most experienced manager predict the total number of labor hours that will be required to complete an upcoming move. This approach has proved useful in the past, but the owner has the business objective of developing a more accurate method of predicting labor hours. In a preliminary effort to provide a more accurate method, the owner has decided to use the number of cubic feet moved and whether there is an elevator in the apartment building as the independent variables and has collected data for 36 moves in which the origin and destination were within the borough of Manhattan in New York City and the travel time was an insignificant portion of the hours worked. The data are organized and stored in. For (a) through (k), do not include an interaction term Moving.
a. State the multiple regression equation for predicting labor hours, using the number of cubic feet moved and whether there is an elevator.
b. Interpret the regression coefficients in (a).
c. Predict the labor hours for moving 500 cubic feet in an apartment building that has an elevator and construct a $95 \%$ confidence interval estimate and a $95 \%$ prediction interval.
d. Perform a residual analysis on the results and determine whether the regression assumptions are valid.
e. Is there a significant relationship between labor hours and the two independent variables (cubic feet moved and whether there is an elevator in the apartment building) at the 0.05 level of significance?
f. At the 0.05 level of significance, determine whether each independent variable makes a contribution to the regression model. Indicate the most appropriate regression model for this set of data.
g. Construct a $95 \%$ confidence interval estimate of the population slope for the relationship between labor hours and cubic feet moved.
h. Construct a $95 \%$ confidence interval estimate for the relationship between labor hours and the presence of an elevator.
i. Compute and interpret the adjusted $r^2$.
j. Compute the coefficients of partial determination and interpret their meaning.
k. What assumption do you need to make about the slope of labor hours with cubic feet moved?
1. Add an interaction term to the model and, at the 0.05 level of significance, determine whether it makes a significant contribution to the model.
m. On the basis of the results of (f) and (1), which model is most appropriate? Explain.

Basic Business Statistics: Concepts and Applications

Mark L. Berenson,… 12th Edition

Chapter 14, Problem 43

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Step 1: **State the multiple regression equation** The multiple regression equation can be stated as: \[ Y = \beta_0 + \beta_1X_1 + \beta_2X_2 + \epsilon \] where $ Y $ is the total labor hours, $ X_1 $ is the number of cubic feet moved, $ X_2 $ is a binary Show more…

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The owner of a moving company typically has his most experienced manager predict the total number of labor hours that will be required to complete an upcoming move. This approach has proved useful in the past, but the owner has the business objective of developing a more accurate method of predicting labor hours. In a preliminary effort to provide a more accurate method, the owner has decided to use the number of cubic feet moved and whether there is an elevator in the apartment building as the independent variables and has collected data for 36 moves in which the origin and destination were within the borough of Manhattan in New York City and the travel time was an insignificant portion of the hours worked. The data are organized and stored in. For (a) through (k), do not include an interaction term Moving. a. State the multiple regression equation for predicting labor hours, using the number of cubic feet moved and whether there is an elevator. b. Interpret the regression coefficients in (a). c. Predict the labor hours for moving 500 cubic feet in an apartment building that has an elevator and construct a $95 \%$ confidence interval estimate and a $95 \%$ prediction interval. d. Perform a residual analysis on the results and determine whether the regression assumptions are valid. e. Is there a significant relationship between labor hours and the two independent variables (cubic feet moved and whether there is an elevator in the apartment building) at the 0.05 level of significance? f. At the 0.05 level of significance, determine whether each independent variable makes a contribution to the regression model. Indicate the most appropriate regression model for this set of data. g. Construct a $95 \%$ confidence interval estimate of the population slope for the relationship between labor hours and cubic feet moved. h. Construct a $95 \%$ confidence interval estimate for the relationship between labor hours and the presence of an elevator. i. Compute and interpret the adjusted $r^2$. j. Compute the coefficients of partial determination and interpret their meaning. k. What assumption do you need to make about the slope of labor hours with cubic feet moved? 1. Add an interaction term to the model and, at the 0.05 level of significance, determine whether it makes a significant contribution to the model. m. On the basis of the results of (f) and (1), which model is most appropriate? Explain.

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Key Concepts

Residual Analysis

Residual analysis involves examining the differences between observed and predicted values to assess the validity of the regression assumptions, such as linearity, independence, homoscedasticity, and normality of errors. It helps identify outliers, model misspecification, or other violations that may affect the reliability of the regression results.

Multiple Linear Regression

This statistical method models the relationship between one dependent variable and two or more independent variables by fitting a linear equation to observed data. It is used for prediction and inference, allowing analysts to understand how changes in predictor variables are associated with changes in the response variable.

Regression Coefficients Interpretation

Each coefficient in a multiple regression represents the expected change in the dependent variable for a one-unit change in an independent variable, holding all other variables constant. The intercept represents the expected value of the dependent variable when all predictors are zero, and slope coefficients describe how the dependent variable changes with each predictor.

Dummy Variables in Regression

Dummy variables are used to represent categorical data in regression models. They allow the inclusion of qualitative factors by converting categories into binary indicators, where each is coded typically as 0 or 1, enabling the model to estimate different intercepts or slopes for different groups.

Confidence and Prediction Intervals

Confidence intervals provide a range where the true mean of the dependent variable is likely to fall for a given set of predictor values, reflecting the uncertainty in parameter estimates. Prediction intervals estimate the range where a new observation is likely to fall, accounting for both the uncertainty of the mean estimate and the variability of individual responses.

Hypothesis Testing in Regression

This concept involves using statistical tests, such as t-tests for individual regression coefficients and F-tests for the overall model, to determine whether the relationships observed between the dependent and independent variables are statistically significant. It helps in deciding if specific predictors have a meaningful contribution to the model.

Adjusted R-Squared

Adjusted R-squared measures the proportion of variation in the dependent variable explained by the regression model while adjusting for the number of predictors. It provides a more accurate measure of model fit than the regular R-squared, especially when comparing models with different numbers of independent variables.

Partial Coefficient of Determination

This coefficient measures the unique contribution of an individual predictor to the overall model fit, quantifying how much additional variance in the dependent variable is explained by that predictor, after accounting for the other variables in the model.

Interaction Terms in Regression

Interaction terms are used to assess whether the effect of one independent variable on the dependent variable depends on the level of another independent variable. Including interaction terms in a model allows for more complex relationships and can reveal if the relationship between predictors and the outcome is not simply additive.

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