On a peer-to-peer (P2P) lending website, borrowers complete an approval scoring form that lenders use to assess creditworthiness. Lenders generally believe that borrowers who score at least73do not default on loans. Consequently, borrowers are ratedAif their overall score is at least73, otherwise they are ratedB.

A reasonably large sample of real borrower data was collected:

1. among those thatdid not defaulton their loans, initial approval scores were normally distributed with a mean of 78.3 and a standard deviation of 7.5.
3. among those thatdefaultedon their loans, initial approval scores were normally distributed with a mean of 66.5 and a standard deviation of 8.9.

Report each answer as a decimal (not percent) accurate to at least 4 decimal places. Answers from software or from roundedz-scores (to 2 decimal places) are accepted.

What proportion of borrowers that:

a)defaultedwere initially ratedA?

b)did not defaultwere initially ratedB?

c)defaultedwere miscategorized initially?

d)did not defaultwere miscategorized initially?

Among those thatdefaulted, what is the probability that a borrower:

a) scored above 86.3?

b) scored below 60 or above 91?

c) was ratedBand scored above 45.4?

d) was ratedAand scored below 86.3?

e) was ratedAgiven scored above 66.5?

f) scored below 66.5 given ratedB?