Profitability remains a challenge for banks and thrifts with less than $$2$ billion of assets. The business problem facing a bank analyst relates to the factors that affect return on assets $($ROA$),$ an indicator of how profitable a company is relative to its total assets. Data collected from a sample of $20$ community banks include the ROA $(\%),$ the efficiency ratio $(\%),$ as a measure of bank productivity $($the lower the efficiency ratio, the better$),$ and total risk$-$based capital $(\%),$ as a measure of capital adequacy. Complete parts $($a$)$ through $($c$)$ below.

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a$.$ State the multiple regression equation.

Let $x_{1i}$ represent the efficiency ratio $(\%)$ and let $x_{2i}$ represent the total risk$-$based capital $(\%).$

$($Round the constant to two decimal places as needed. Round the coefficients to four decimal plac

b$.$ Interpret the meaning of the slopes, $b_1$ and $b_2,$ in this problem. Choose the correct answer belo

A$.$ For a given Risk$-$Based Capital, for each increase of $1\%$ in the Efficiency Ratio, the ROA is in Risk$-$Based Capital, the ROA is estimated to increase by $b_2\%.$

B$.$ For each increase of $1\%$ in the ROA, the Efficiency Ratio is estimated to increase by $b_1\%$

C$.$ For each increase of $1\%$ in both the Efficiency Ratio and the Risk$-$Based Capital, the ROA

D$.$ The slopes, $b_1$ and $b_2,$ cannot be interpreted individually.

c$.$ Predict the mean ROA when the efficiency ratio is $50\%$ and the total risk$-$based capital is $25\%.$

$\%$

$($Round to two decimal places as needed.$)$

Distribution Costs

$\backslash$table$[[$ROA $(\%),\backslash$table$[[$Efficiency$],[$Ratio $(\%)]],\backslash$table$[[$Total Risk$-$Based$],[$Capital $(\%)]]],[0\mathrm{.}60,63\mathrm{.}31,13\mathrm{.}94],[1\mathrm{.}10,79\mathrm{.}47,10\mathrm{.}53],[0\mathrm{.}86,64\mathrm{.}23,14\mathrm{.}20],[0\mathrm{.}96,49\mathrm{.}12,12\mathrm{.}91],[0\mathrm{.}54,74\mathrm{.}69,15\mathrm{.}33],[0\mathrm{.}84,62\mathrm{.}66,20\mathrm{.}53],[1\mathrm{.}06,57\mathrm{.}09,14\mathrm{.}62],[1\mathrm{.}76,34\mathrm{.}58,15\mathrm{.}73],[1\mathrm{.}08,64\mathrm{.}53,20\mathrm{.}72],[1\mathrm{.}07,64\mathrm{.}82,13\mathrm{.}86],[0\mathrm{.}81,69\mathrm{.}59,13\mathrm{.}27],[1\mathrm{.}15,63\mathrm{.}52,15\mathrm{.}18],[1\mathrm{.}02,57\mathrm{.}76,13\mathrm{.}54],[1\mathrm{.}06,59\mathrm{.}37,14\mathrm{.}06],[1\mathrm{.}17,62\mathrm{.}47,11\mathrm{.}67],[1\mathrm{.}14,74\mathrm{.}06,14\mathrm{.}70],[1\mathrm{.}14,56\mathrm{.}32,16\mathrm{.}10],[1\mathrm{.}04,69\mathrm{.}48,18\mathrm{.}47],[0\mathrm{.}82,50\mathrm{.}93,13\mathrm{.}39],[0\mathrm{.}81,77\mathrm{.}32,22\mathrm{.}86]]$