A Manufacturer of Boiler Drums Wants to Use Regression to Predict

question 23

Multiple Choice

A manufacturer of boiler drums wants to use regression to predict the number of man-hours needed to erect drums in the future. The manufacturer collected a random sample of 35 boilers and measured the following two variables: MANHRS: $\quad y =$ Number of man-hours required to erect the drum
PRESSURE: $\quad x =$ Boiler design pressure (pounds per square inch, i.e., $\mathrm { psi }$ )
The simple linear model $E ( y ) = \beta _ { 1 } + \beta _ { 1 } x$ was fit to the data. A printout for the analysis appears below:
UNWEIGHTED LEAST SQUARES LINEAR REGRESSION OF MANHRS
$\begin{array}{c|c|c|c|c}\text { PREDICTOR } & & & & \\\text { VARIABLES } & \text { COEFFICIENT } & \text { STD ERROR } & \text { STUDENT'S T } & \text { P } \\\hline \text { CONSTANT } & 1.88059 & 0.58380 & 3.22 & 0.0028 \\\text { PRESSURE } & 0.00321 & 0.00163 & 2.17 & 0.0300\end{array}$

$\begin{array} { l l l l } \text { R-SQUARED } & 0.4342 & \text { RESID. MEAN SQUARE (MSE) } & 4.25460 \\ \text { ADJUSTED R-SQUARED } & 0.4176 & \text { STANDARD DEVIATION } & 2.06267 \end{array}$

$\begin{array}{l|r|c|c|c|c}\text { SOURCE } & \text { DF } & {\text { SS }} & \text { MS } & \mathrm{F} & \mathrm{P} \\\hline \text { REGRESSION } & 1 & 111.008 & 111.008 & 5.19 & 0.0300 \\\text { RESIDUAL } & 34 & 144.656 & 4.25160 & & \\\text { TOTAL } & 35 & 255.665 & & &\end{array}$

Give a practical interpretation of the coefficient of determination, $r ^ { 2 }$ .

• A) About 43% of the sample variation in number of man-hours can be explained by the simple linear model.
• B) We are 43% confident that the design pressure will be a useful predictor of number of man-hours required to build a steam drum.
• C) Approximately 95% of the actual man-hours required to build a drum will fall within 43 hours of their predicted values.
• D) About 2.06% of the sample variation in number of man-hours can be explained by the simple linear model.

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Definitions:

Type I Error

The improper denial of a genuine null hypothesis, also identified as a "false positive."

Randomly Occurring

Events or phenomena that happen without a predictable pattern, order, or reason.

Production Process

A series of steps, operations, or stages designed to produce a product or achieve a specific outcome in a manufacturing or production environment.

Statistical Control

The process of statistically adjusting variables or data to isolate the effect of one variable from others in an analysis.