The average college graduate has a student loan debt of 2523

The average college graduate has a student loan debt of $25,236. Assume the population standard deviation is $5,000. Suppose you take a random sample of 50 college graduates.

16. To 4 decimal places, what is the probability that the sample mean debt is more than $26,000?

17. To 4 decimal places, what is the probability that the sample mean debt is less than $25,000?

18. To 4 decimal places, what is the probability that the sample mean debt is within $1,000 of the population mean (i.e., from $1,000 below, to $1,000 above?

Solution

Normal Distribution
Mean ( u ) =25236
Standard Deviation ( sd )=5000
Normal Distribution = Z= X- u / sd ~ N(0,1)                  
a)              
P(X > 26000) = (26000-25236)/5000
= 764/5000 = 0.1528
= P ( Z >0.153) From Standard Normal Table
= 0.4393                  

b)
P(X < 25000) = (25000-25236)/5000
= -236/5000= -0.0472
= P ( Z <-0.0472) From Standard Normal Table
= 0.4812  
c)
To find P(a < = Z < = b) = F(b) - F(a)
P(X < 24236) = (24236-25236)/5000
= -1000/5000 = -0.2
= P ( Z <-0.2) From Standard Normal Table
= 0.42074
P(X < 26236) = (26236-25236)/5000
= 1000/5000 = 0.2
= P ( Z <0.2) From Standard Normal Table
= 0.57926
P(24236 < X < 26236) = 0.57926-0.42074 = 0.1585