Cost Analysis $1$: Flagstone Piston Corporation's products are replacement automobile parts. Flagstone had experienced a rather stable growth in sales in recent years and had been required to increase its capacity regularly. The executive officers had been asked by Flagstone's directors to formulate a proposal for expansion and price adjustments.

Quarterly data were obtained on manufacturing costs per unit $($in hundreds of dollars$),$ production level $($a proportion of the total capacity$),$ and the index of direct material and direct labor costs for a five$-$year period $($These data are in the worksheet entitled PISTON$5).$

Begin by fitting the linear model to predict the average cost per unit $($COST$)$ from production level $($x$1=$ PROD$)$ and the index of direct material and direct labor costs $($x$2=$ INDEX$).$

a$)$ State the linear model equation.

$=0+1$x$1+2$

$($

$1$

x$2$

$)$

$=0+1$ln$($x$1)+2$x$2$

$=0+1$x$1+2$ln$($x$2)$

$=0+1($

$1$

x$1$

$)+2$x$2$

$=0+1$x$1+2$x$2$

Correct: Your answer is correct.

However, there is previous research to suggest that the relationship between cost and production level may best be modeled by using the reciprocal of production level in the regression model.

b$)$ State the transformed model equation $($i$.$e$.,$ the reciprocal model equation$).$

$=0+1$x$1+2($

$1$

x$2$

$)$

$=0+1$ln$($x$1)+2$x$2$

$=0+1($

$1$

x$1$

$)+2$x$2$

$=0+1$x$1+2$ln$($x$2)$

$=0+1$x$1+2$x$2$

Correct: Your answer is correct.

c$)$ How does the reciprocal model compare with the original, linear model? In answering this question, fit both models and use the R$2$ values and the standard error of the regression in the comparisons. Provide summary measures about the two models in the table below. $($Enter the R$2$ values as percents to two decimal places and round your standard errors to three decimal places.$)$

Model R$2$ R$2$adj se

Linear Model

$91\mathrm{.}93$

Correct: Your answer is correct.

$\%$

$90\mathrm{.}98$

Correct: Your answer is correct.

$\%$

$0\mathrm{.}228$

Correct: Your answer is correct.

Reciprocal Model

$95\mathrm{.}54$

Correct: Your answer is correct.

$\%$

$95\mathrm{.}01$

Correct: Your answer is correct.

$\%$

$0\mathrm{.}169$

Correct: Your answer is correct.

d$)$ Which is the better fitting model?

The reciprocal production model obtained a lower R$2$ and a larger standard error, and therefore appears to be an improvement.

The reciprocal production model obtained a higher R$2$ and a smaller standard error, and therefore appears to be an improvement.

The linear model obtained a higher R$2$ and a smaller standard error, and therefore appears to be superior.

The linear model obtained a lower R$2$ and a larger standard error, and therefore appears to be superior.

Correct: Your answer is correct.

Regardless of your decision above, use the Reciprocal Model to answer ALL of the questions below. Also, be sure to compute these values based on the UNROUNDED values in EXCEL.

e$)$ Predict the cost per unit for a quarter with a production level of $75\%$ and an index of $90.($Enter your answers to two decimal places.$)$

$

$176\mathrm{.}43$

Incorrect: Your answer is incorrect.

f$)$ Predict the cost per unit for a quarter with a production level of $85\%$ and an index of $95.($Enter your answers to two decimal places.$)$

$

$198\mathrm{.}07$

Incorrect: Your answer is incorrect.$\text{'}$