Question

An aircraft company wanted to predict the number of worker-hours necessary to finish the design of a new plane. Relevant explanatory variables were thought to be the plane's top speed, its weight, and the number of parts it had in common with other models built by the company. A sample of 27 of the company's planes was taken, and the following model was estimated: $$ y_i=\beta_0+\beta_1 x_{1 i}+\beta_2 x_{2 i}+\beta_3 x_{3 i}+\varepsilon_i $$ where $y_i=$ design effort, in millions of worker-hours $x_{1 i}=$ plane's top speed, in miles per hour $x_{2 i}=$ plane's weight, in tons $x_{3 i}=$ percentage number of parts in common with other models The estimated regression coefficients were as follows: $$ b_0=2 \quad b_1=0.661 \quad b_2=0.065 \quad b_3=-0.018 $$ Interpret these estimates.

   An aircraft company wanted to predict the number of worker-hours necessary to finish the design of a new plane. Relevant explanatory variables were thought to be the plane's top speed, its weight, and the number of parts it had in common with other models built by the company. A sample of 27 of the company's planes was taken, and the following model was estimated:
$$
y_i=\beta_0+\beta_1 x_{1 i}+\beta_2 x_{2 i}+\beta_3 x_{3 i}+\varepsilon_i
$$
where
$y_i=$ design effort, in millions of worker-hours
$x_{1 i}=$ plane's top speed, in miles per hour
$x_{2 i}=$ plane's weight, in tons
$x_{3 i}=$ percentage number of parts in common with other models

The estimated regression coefficients were as follows:
$$
b_0=2 \quad b_1=0.661 \quad b_2=0.065 \quad b_3=-0.018
$$
Interpret these estimates.
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Statistics for Business and Economics: Global Edition
Statistics for Business and Economics: Global Edition
Newbold P., Carlson… 8th Edition
Chapter 12, Problem 6 ↓

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In practical terms, this scenario is hypothetical as a plane cannot have zero speed, zero weight, or zero parts. Thus, the intercept here is more of a statistical artifact rather than a practically interpretable figure.  Show more…

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An aircraft company wanted to predict the number of worker-hours necessary to finish the design of a new plane. Relevant explanatory variables were thought to be the plane's top speed, its weight, and the number of parts it had in common with other models built by the company. A sample of 27 of the company's planes was taken, and the following model was estimated: $$ y_i=\beta_0+\beta_1 x_{1 i}+\beta_2 x_{2 i}+\beta_3 x_{3 i}+\varepsilon_i $$ where $y_i=$ design effort, in millions of worker-hours $x_{1 i}=$ plane's top speed, in miles per hour $x_{2 i}=$ plane's weight, in tons $x_{3 i}=$ percentage number of parts in common with other models The estimated regression coefficients were as follows: $$ b_0=2 \quad b_1=0.661 \quad b_2=0.065 \quad b_3=-0.018 $$ Interpret these estimates.
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Key Concepts

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Multiple Linear Regression
This is a statistical technique used to model the relationship between a single dependent variable and several independent variables. It estimates the effect of each independent variable on the dependent outcome, holding all other variables constant, which helps in understanding the relative influence of each predictor.
Intercept Interpretation
In a regression model, the intercept represents the predicted value of the dependent variable when all independent variables are set to zero. Its interpretation should consider whether zero is a meaningful value for the predictors, as it may not always correspond to a real or expected scenario.
Slope Coefficient Interpretation
Each slope coefficient in a multiple regression model reflects the expected change in the dependent variable resulting from a one-unit increase in the corresponding independent variable, with all other variables held constant. A positive slope indicates an increase in the dependent variable, while a negative slope indicates a decrease.
Ceteris Paribus Assumption
This assumption underpins the interpretation of regression coefficients, meaning 'all else being equal.' It clarifies that the effect of a one-unit change in an independent variable on the dependent variable is evaluated while keeping the other predictors fixed, which simplifies the analysis of complex relationships.

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In a study of the performance of a new engine design, the weight of 22 aircrafts (in tons) and the top speed (in mph) were recorded. A regression line was generated and shown to be an appropriate description of the relationship. The results of the regression analysis are below. Depend Variable: Top Speed Variable Constant Weight Coefficient 11.6559 3.47812 s.e. of Coeff 0.3153 0.294 t-ratio 37 11.8 prob ≤ 0.0001 ≤ 0.0001 R squared = 87.5% R squared (adjusted) = 86.9% s = 0.6174 with 22 - 2 = 20 degrees of freedom Part A: What is the LSRL based on the analysis provided? Make sure to identify what the variables represent in the context of the problem. (4 points) Part B: What is the predicted value for the top speed of an aircraft if its weight is 100 tons? Show your work. (6 points)

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