An aircraft company wanted to predict the number of...

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: $$ y=\beta_0+\beta_1 x_1+\beta_2 x_2+\beta_3 x_3+\varepsilon $$ where $y=$ design effort, in millions of worker-hours $x_1=$ plane's top speed, in miles per hour $x_2=$ plane's weight, in tons $x_3=$ percentage of parts in common with other models The estimated regression coefficients were as follows: $$ b_1=0.661 \quad b_2=0.065 \quad b_3=-0.018 $$ The estimated standard errors were as follows: $$ s_{b_1}=0.099 \quad \mathrm{~s}_{b_2}=0.032 \quad \mathrm{~s}_{b_3}=0.0023 $$ a. Find $90 \%$ and $95 \%$ confidence intervals for $\beta_1$. b. Find $95 \%$ and $99 \%$ confidence intervals for $\beta_2$. c. Test against a two-sided alternative the null hypothesis that, all else being equal, the plane's weight has no linear influence on its design effort. d. The error sum of squares for this regression was 0.332 . Using the same data, a simple linear regression of design effort on the percentage of common parts was fitted, yielding an error sum of squares of 3.311. Test, at the $1 \%$ level, the null hypothesis that, taken together, the variable's top speed and weight contribute nothing in a linear sense to explaining the changes in the variable, design effort, given that the variable percentage of common parts is also used as an explanatory variable.

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:
$$
y=\beta_0+\beta_1 x_1+\beta_2 x_2+\beta_3 x_3+\varepsilon
$$
where
$y=$ design effort, in millions of worker-hours
$x_1=$ plane's top speed, in miles per hour
$x_2=$ plane's weight, in tons
$x_3=$ percentage of parts in common with other models

The estimated regression coefficients were as follows:
$$
b_1=0.661 \quad b_2=0.065 \quad b_3=-0.018
$$
The estimated standard errors were as follows:
$$
s_{b_1}=0.099 \quad \mathrm{~s}_{b_2}=0.032 \quad \mathrm{~s}_{b_3}=0.0023
$$
a. Find $90 \%$ and $95 \%$ confidence intervals for $\beta_1$.
b. Find $95 \%$ and $99 \%$ confidence intervals for $\beta_2$.
c. Test against a two-sided alternative the null hypothesis that, all else being equal, the plane's weight has no linear influence on its design effort.
d. The error sum of squares for this regression was 0.332 . Using the same data, a simple linear regression of design effort on the percentage of common parts was fitted, yielding an error sum of squares of 3.311. Test, at the $1 \%$ level, the null hypothesis that, taken together, the variable's top speed and weight contribute nothing in a linear sense to explaining the changes in the variable, design effort, given that the variable percentage of common parts is also used as an explanatory variable.

Key Concepts

Hypothesis Testing for Individual Regression Coefficients

This involves testing whether a specific predictor variable has a significant impact on the response variable by establishing a null hypothesis (typically that the coefficient is zero) and using a t-test. If the calculated t-statistic exceeds the critical t-value at a chosen significance level, the null hypothesis is rejected, indicating that the predictor has a statistically significant effect.

F-Test for Nested Models

An F-test is used to assess whether a group of predictors together significantly improves the model fit compared to a restricted model that excludes them. This test compares the reduction in the error sum of squares when the predictors are included, providing a joint significance test that identifies whether these variables contribute meaningful information beyond another established predictor.

Confidence Intervals for Regression Coefficients

A confidence interval provides a range within which the true value of a regression coefficient is likely to lie, with a given level of confidence. It is computed using the estimated coefficient, its standard error, and the appropriate critical value from the t-distribution, reflecting the uncertainty inherent in the estimation process.

Standard Errors and t-Distribution

Standard errors measure the variability of the estimated regression coefficients and are essential for quantifying the precision of these estimates. Under the assumption of normally distributed errors, the ratio of the estimated coefficient to its standard error follows a t-distribution, which is used for constructing confidence intervals and conducting hypothesis tests.

Multiple Linear Regression

A statistical method that models the relationship between one dependent variable and several independent variables. It estimates the impact of predictors on the outcome using coefficients, helping to explain variation in the response. The model is typically assessed using metrics and inference procedures to determine the reliability and significance of the predictors.

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