Econ 306 Hw Solutions
100 Points Total
Answer the following questions as well as you can. LATE HOMEWORKS ARE NEVER ACCEPTED. You may meet/consult with colleagues in the class. But the assignment you turn in needs to be your own work. You should show some (though not necessarily every bit) of work for any substantial calculations.
1. (Each part 5 points) Suppose . That is, X has a normal distribution with μ=30 and σ2=144.
1a. Find a transformation of that will give it a mean of zero and a variance of one (ie., standardize ).
Let the transformed variable be named Z. We desire μz=0, σ2z=1. This means 0=a+b μX and b2 σ2X=1. One solution to this system of equations is b=1/12 and a=-5/2.
Of course, if you recognized the fact that our standard Z-transformation accomplishes precisely this, you could write the transformation as .
1b. Find the probability that . Convert X into a standard normal via Z= . For X=18, Z=-1. For X=36, Z=.5. The probability that X is between 18 and 36 is thus equivalent to the probability of Z between -1 and .5. The latter term is F(.5)-F(-1)=.6915-.1587=.5328
1c. Supposing 5X, find the mean of . This is actually easier than 1a or 1b. μY =a+b μX=10+5*30=160
1d. Find the variance of . σ2Y =b2 σ2X=52*144=3600
2. (Each part 4 points) A bank has been receiving complaints from real estate agents that their customers have been waiting too long for mortgage confirmations. The bank prides itself on its mortgage application process and decides to investigate the claims. The bank manager takes a random sample of 20 customers whose mortgage applications have been processed in the last 6 months and finds the following wait times (in days):
5, 7, 22, 4, 12, 9, 9, 14, 3, 6, 5, 8, 10, 17, 12, 10, 9, 4, 3, 13
Assume that the random variable measures the number of days a customer waits for mortgage processing at this bank, and assume that is normally distributed.
2a. Find the sample mean of this
Answer the following questions as well as you can. LATE HOMEWORKS ARE NEVER ACCEPTED. You may meet/consult with colleagues in the class. But the assignment you turn in needs to be your own work. You should show some (though not necessarily every bit) of work for any substantial calculations.
1. (Each part 5 points) Suppose . That is, X has a normal distribution with μ=30 and σ2=144.
1a. Find a transformation of that will give it a mean of zero and a variance of one (ie., standardize ).
Let the transformed variable be named Z. We desire μz=0, σ2z=1. This means 0=a+b μX and b2 σ2X=1. One solution to this system of equations is b=1/12 and a=-5/2.
Of course, if you recognized the fact that our standard Z-transformation accomplishes precisely this, you could write the transformation as .
1b. Find the probability that . Convert X into a standard normal via Z= . For X=18, Z=-1. For X=36, Z=.5. The probability that X is between 18 and 36 is thus equivalent to the probability of Z between -1 and .5. The latter term is F(.5)-F(-1)=.6915-.1587=.5328
1c. Supposing 5X, find the mean of . This is actually easier than 1a or 1b. μY =a+b μX=10+5*30=160
1d. Find the variance of . σ2Y =b2 σ2X=52*144=3600
2. (Each part 4 points) A bank has been receiving complaints from real estate agents that their customers have been waiting too long for mortgage confirmations. The bank prides itself on its mortgage application process and decides to investigate the claims. The bank manager takes a random sample of 20 customers whose mortgage applications have been processed in the last 6 months and finds the following wait times (in days):
5, 7, 22, 4, 12, 9, 9, 14, 3, 6, 5, 8, 10, 17, 12, 10, 9, 4, 3, 13
Assume that the random variable measures the number of days a customer waits for mortgage processing at this bank, and assume that is normally distributed.
2a. Find the sample mean of this