# Confidence interval of bad debt ratio for financial institutions

The bad debt ratio for a financial institution is defined to be the dollar value of loans defaulted divided by the total dollar value of all loans made. A random sample of 6 Ontario banks is selected and that the bad debt ratios (in percentages) for these banks are:

7 4 6 7 5 8

Compute and interpret a 95% confidence interval for the mean bad debt ratio. What needs to be true in order for this interval and interpretation to be valid?

My work:

I started off by finding the mean: (7+4+6+7+5+8)/6 = 6.167, but I am confused as to how to find the variance of 2.167.

I continued using the answer for the variance and did: 6.167 + 1.96*(1.4721/sqrt6) , 6.167 - 1.96*(1.4721/sqrt6), and I got the answer of [4.987, 7.347], but the answer says [4.62,7.71].

Any help as to how to find the variance and where I went wrong in my confidence interval is very appreciated! Thank you!

The variance has presumably been calculated as $$\frac{\left(7-\frac{37}{6}\right)^2+\left(4-\frac{37}{6}\right)^2+\left(6-\frac{37}{6}\right)^2+\left(7-\frac{37}{6}\right)^2+\left(5-\frac{37}{6}\right)^2+\left(8-\frac{37}{6}\right)^2}{6-1}$$ with division by $6-1$ rather than $6$ to give an unbiased estimate of the population estimate.
With such as small sample, even if you assume a normal distribution for the population, you should probably be using a Student's $t$-distribution for the confidence interval.