# Question: An aircraft company wanted to predict the number of worker hours

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 = β0 + β1x1 + β2x2 + β3x3 + ε

where

y = design effort, in millions of worker-hours

x1 = plane’s top speed, in miles per hour

x2 = plane’s weight, in tons

x3 = percentage of parts in common with other models

The estimated regression coefficients were as follows:

b1 = 0.661 b2 = 0.065 b3 = -0.018

The estimated standard errors were as follows:

sb1 = 0.099 sb2 = 0.032 sb3 = 0.0023

a. Find 90% and 95% confidence intervals for β1.

b. Find 95% and 99% confidence intervals for β2.

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.

y = β0 + β1x1 + β2x2 + β3x3 + ε

where

y = design effort, in millions of worker-hours

x1 = plane’s top speed, in miles per hour

x2 = plane’s weight, in tons

x3 = percentage of parts in common with other models

The estimated regression coefficients were as follows:

b1 = 0.661 b2 = 0.065 b3 = -0.018

The estimated standard errors were as follows:

sb1 = 0.099 sb2 = 0.032 sb3 = 0.0023

a. Find 90% and 95% confidence intervals for β1.

b. Find 95% and 99% confidence intervals for β2.

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.

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