Question

Student loans: In an article from September 13, 2010, titled “Student Loan Default Rates Increase,” the website ed.gov analyzes trends in default rates for student loans over time. For the year 2008, the default rate for graduates was 7%. Suppose that we select random samples from this population of graduates from 2008. For each sample we calculate the proportion who defaulted on their loans. The sample size is 200.

A. check if normal model applies in this situation

B. State the parameter given in the problem

C. Find the standard distribution of the sample distribution (standard error)

D. Find the probability that in a random sample of 200 graduates, the default rate is more than 7.5%. Is this unusual?

Answer 1

A) The conditions for applying Normal model is

n is large (n>30)

np 10 and n(1-p) 10

Here n = 200 is large

np = 200*0.07 = 14 > 10

n(1-p) = 200* (1-0.07 ) = 186 >10

Thus we can apply Normal model.

B) Parameter is the population proportion of graduates who defaulted on their loans, p =0.07

C) The sampling distribution of sample proportion , follow Normal with mean = p = 0.07

and standard error =

D) To find

We know that

= P(z > 0.28 )

= 0.3897 ( from z table )

Probability that a random sample of 200 graduates , default rate more than 7.5% is 0.3897

Since Probability is greater than 0.05

It is not unusual that a random sample of 200 graduates , default rate is more than 7.5% given that population default rate is 7%.