Failure to meet payments on student loans guaranteed by the

Failure to meet payments on student loans guaranteed by the US government has been a major problem for both banks and the government. Approximately 50 percent of all student loans guaranteed by the government are in default. A random sample of 350 loans to college students in one region of the United States indicates that 147 are in default. Calculate the p-value associated with the test statistic, and test the hypothesis that the default rate for the selected region is lower than the national percentage at = .01.

Solution

Set Up Hypothesis
Null, H0:P=0.5
Alternate,Lower than national percentage H1: P<0.5
Test Statistic
No. Of Success chances Observed (x)=147
Number of objects in a sample provided(n)=350
No. Of Success Rate ( P )= x/n = 0.42
Success Probability ( Po )=0.5
Failure Probability ( Qo) = 0.5
we use Test Statistic (Z) for Single Proportion = P-Po/Sqrt(PoQo/n)
Zo=0.42-0.5/(Sqrt(0.25)/350)
Zo =-2.9933
| Zo | =2.9933
Critical Value
The Value of |Z | at LOS 0.01% is 2.33
We got |Zo| =2.993 & | Z | =2.33
Make Decision
Hence Value of | Zo | > | Z | and Here we Reject Ho
P-Value: Left Tail -Ha : ( P < -2.99333 ) = 0.00138
Hence Value of P0.01 > 0.00138,Here we Reject Ho

Lower than national percentage