 # A bank finds that the estimated proportion of clients defaulting on a loan, give emancipezN 2021-09-08 Answered
A bank finds that the estimated proportion of clients defaulting on a loan, given the interest rate is below $$\displaystyle{15}\%$$, is 0.34, They also find that the estimated proportion of clients defaulting on a loan, given the interest is greater than or equal to $$\displaystyle{15}\%$$, is 0.52.
a) What is the odds ratio of defaulting given the interest rate is greater than or equal to $$\displaystyle{15}\%$$ relative to the interest rate is lower than $$\displaystyle{15}\%$$? Interpret this odds ratio.
b) If we were to analyze this data using the following logistic regression model, what are the estimates of $$\displaystyle\beta_{{{0}}}$$ and $$\displaystyle\beta_{{{1}}}$$? Show your work.
$$\displaystyle{l}{g}{\left\lbrace{\left({\frac{{{p}}}{{{1}-{p}}}}\right)}\right\rbrace}=\beta_{{{0}}}+\beta_{{{1}}}{x}$$
Where p is the probability of defaulting on a loan and x is an indicator variable that is 1 when the interest rate is greater than or equal to $$\displaystyle{15}\%$$ and 0 when the interest rate is less than $$\displaystyle{15}\%$$.

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SOLUTION: we translate data into tabular formate.
$$\begin{array}{|c|c|}\hline &&Cases&Contol\\ \hline &&Defaulting&non-Defaulting\\ \hline exposed&<15\%&0.34&0.66\\ \hline unexposed&\geq15\%&0.52&0.48\\ \hline\end{array}$$
a) What is the odds ratio of defaulting given the interest rate is greater than or equal to $$\displaystyle{15}\%$$ relative to the interest rate is lower than $$\displaystyle{15}\%$$? Interpret this odds ratio.
ANS: odds of defaulting given interest rate $$\displaystyle{15}\geq{15}\%$$ is $$\displaystyle{o}{d}{d}{1}={\left({\frac{{{0.52}}}{{{0.48}}}}\right)}={1.083}$$
odds of defaulting given interest rate $$\displaystyle{<}{15}\%$$ is $$\displaystyle{o}{d}{d}{1}={\left({\frac{{{0.34}}}{{{0.66}}}}\right)}={0.5151}$$
SO odds ratio is given by,
addratio=$$\displaystyle{\frac{{{o}{d}{d}{s}{1}}}{{{o}{d}{d}{s}{20}}}}={\frac{{{1.083}}}{{{0.5151}}}}={2.1025}$$
Interpretation: odds of defaulting given interest rate $$\displaystyle{15}\geq{15}\%$$ is greater than odds of defaulting given interest rate $$\displaystyle{<}{15}\%$$, that is high intereset rate $$\displaystyle{15}\geq{15}\%$$ may be a risk factor for defualting a loan.
b) If we were to analyze this data using the following logistic regression model, what are the estimates of $$\displaystyle\beta_{{{0}}}$$ and $$\displaystyle\beta_{{{1}}}$$? Show your work.
$$\displaystyle{l}{g}{\left\lbrace{\left({\frac{{{p}}}{{{1}-{p}}}}\right)}\right\rbrace}=\beta_{{{0}}}+\beta_{{{1}}}{x}$$
ANS:
where bl and bo are $$\displaystyle{b}{1}={\log{}}$$ (base catergory) and $$\displaystyle{b}{1}={\log{}}$$ (odd ratio)
$$\displaystyle{b}{o}={\log{}}$$ (odds of defaulting given interest rate $$\displaystyle{15}\geq{15}\%={\log{{\left({1.083}\right)}}}={0.03462}$$
$$\displaystyle{b}{1}={\log{}}$$ (oddratio of defaulting given interest rate $$\displaystyle{15}\geq{15}\%$$ relative to the defaulting given interest rate $$\displaystyle{<}{15}\%={\log{{\left({2.1025}\right)}}}$$