Question: An architect claims that the cost of constructing an office building is related to the area of the floor space in the building according to the quadratic regression model

where \(Y\) is the cost per square foot in dollars and \(X\) is the floor space in \(100,000 \mathrm{~s}\) of square feet. The data are as follows:

\(\begin{array}{lllllllllll}Y & 16 & 19 & 22 & 26 & 28 & 29 & 30 & 33 & 36 & 40 \\ X & 2.6 & 3.4 & 4.3 & 4.5 & 5.0 & 6.2 & 6.8 & 7.2 & 8.4 & 9.7\end{array}\)

1. Plot the data on a scatter diagram. Does a linear or quadratic relationship seem to describe the relationship between \(Y\) and \(X\) better?
2. Estimate the linear regression model, and find the adjusted \(R^{2}\).
3. Estimate the quadratic regression model, and find the adjusted \(R^{2}\).
4. Plot the estimated equations on a scatter diagram.
5. Let \(\alpha=.05 .\) Test \(H_{0}: \beta_{2}=0\) against \(H_{1}: \beta_{2}>0\).
6. Predict the cost of constructing a building that is 200 feet by 200 feet and has six floors. Use the equation estimated in part \(\mathbf{b}\) and then the estimated equation in part c. Which prediction do you think is better?

For part (e), test against \({{\beta }_{2}}\ne 0\).

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