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.

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 places as needed.$)$

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

A$.$ For a given Risk$-$Based Capital, for each increase of $1\%$ in the Efficiency Ratio, the ROA is estimated to increase by 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 $b_1\%$ and the Risk$-$Based Capite

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

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

c$.$ Predict the mean ROA when the efficiency ratio is $50\%$ d the total risk$-$based capital is $20\%.$

$\%$

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

Distribution Costs

$\backslash$table$[[$ROA $(\%),\backslash$table$[[$Efficiency$],[$Ratio $(\%)]],\backslash$table$[[$Total Risk$-$Based$],[$Capital $(\%)]]],[0\mathrm{.}95,64\mathrm{.}60,13\mathrm{.}41],[0\mathrm{.}86,64\mathrm{.}23,14\mathrm{.}20],[1\mathrm{.}04,39\mathrm{.}93,17\mathrm{.}04],[1\mathrm{.}55,71\mathrm{.}04,14\mathrm{.}30],[5\mathrm{.}24,82\mathrm{.}22,21\mathrm{.}24],[1\mathrm{.}07,42\mathrm{.}80,13\mathrm{.}83],[3\mathrm{.}40,86\mathrm{.}43,24\mathrm{.}38],[1\mathrm{.}14,56\mathrm{.}32,16\mathrm{.}10],[0\mathrm{.}99,68\mathrm{.}29,13\mathrm{.}38],[0\mathrm{.}93,61\mathrm{.}61,12\mathrm{.}55],[1\mathrm{.}12,62\mathrm{.}30,17\mathrm{.}88],[1\mathrm{.}62,28\mathrm{.}26,20\mathrm{.}28],[0\mathrm{.}90,80\mathrm{.}64,13\mathrm{.}27],[1\mathrm{.}35,51\mathrm{.}73,11\mathrm{.}92],[1\mathrm{.}25,68\mathrm{.}16,13\mathrm{.}57],[1\mathrm{.}01,79\mathrm{.}14,15\mathrm{.}70],[0\mathrm{.}71,51\mathrm{.}25,13\mathrm{.}45],[0\mathrm{.}83,67\mathrm{.}37,15\mathrm{.}76],[1\mathrm{.}13,64\mathrm{.}81,15\mathrm{.}17],[0\mathrm{.}90,65\mathrm{.}46,16\mathrm{.}12]]$