The bad debt ratio for a financial institution is defined to be dollar value of

Question

The bad debt ratio for a financial institution is defined to be dollar value of loans
defaulted divided by the total dollar value of all loans made. Suppose a random sample of
seven Ohio banks is selected and that the bad debt ratios (written as percentages) for
these banks are 7 percent, 4 percent, 6 percent, 7 percent, 5 percent, 4 percent, and 9
percent. Assuming the bad debt ratios are approximately normally distributed with a 95
percent confidence interval for the mean bad debt ratio of all Ohio banks,
a. Demonstrate the calculation of the 95% confidence interval
b. Calculate 99% confidence interval for the population mean debt-to- equity
ratio.

Explanation / Answer

Xi denotes bad debt ratio and xi's are normally distrbuted.

n=7

sample mean = Xbar = (7+4+6+7+5+4+9)/7 = 6

S2 = (1/n-1) * sum((xi-xbar)^2). which is unbiased estimator of population variance.

S2 = 3.3333 , S= 1.8257

Since n=7 (small) and population variance is unknown.

t= (Xbar-mu)/(S/sqrt(n)) ~ tn-1 where mu is the population mean

The 95% confidence interval for population mean mu are

( Xbar-t6,0.025 *(S*sqrt(n) , Xbar + t6,0.025 *(S*sqrt(n))

table value : t6, 0.025= 2.447

Hence 95% confidence interval for population mean bad ratio are

( 6 - ( 2.447 *1.8257/sqrt(6)),  6 + ( 2.447 *1.8257/sqrt(6))

=(4.1761 , 7.8238)

95% percent mean bad debt ratio between 4.1761% to 7.8238%

Table value : t6,0.005 =3.707

The 99% confidence interval for population mean mu debt to equity ratio are

( Xbar-t6,0.005 *(S*sqrt(n) , Xbar + t6,0.005 *(S*sqrt(n))

( 6 - ( 3.707 *1.8257/sqrt(6)),  6 + ( 3.707 *1.8257/sqrt(6))

=(3.2369 , 8.7630)

99% percent mean bad debt ratio between 3.2369% to 8.7630%

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