$10-30$ Cost$-$volume$-$profit and regression analysis. Grant Corporation manufactures a drink bottle, model CL$24.$ During $2020,$ Grant produced $240,000$ bottles at a total cost of $$1,080,000.$ Wren Corporation has offered to supply as many bottles as Grant wants at a cost of $$4\mathrm{.}40$ per bottle. Grant anticipates needing $255,000$ bottles each year for the next few years.

a$.$ What is the average cost of manufacturing a drink bottle in $2020?$ How does it compare to Wren's offer?

b$.$ Can Grant use the answer in requirement $1$ a to determine the cost of manufacturing $255,000$ drink bottles? Explain.

Grant's cost analyst uses annual data from past years to estimate the following regression equation with total manufacturing costs of the drink bottle as the dependent variable and drink bottles produced as the independent variable:

$y=$$$431,000+$$$3\mathrm{.}00x$

During the years used to estimate the regression equation, the production of bottles varied from $229,000$ to $265,000.$ Using this equation, estimate how much it would cost Grant to manufacture $255,000$ drink bottles. How much more or less costly is it to manufacture the bottles than to acquire them from Wren?

$3.$ What other information would you need to be confident that the equation in requirement $2$ accurately predicts the cost of manufacturing drink bottles?