# 2 (A2) MATH 422/522 SPRING 2017 Due: February 7, 2017 Name________________________________________________________________ Q1. [1+2+1] In a certain bank, there are 15000 deposit accounts. A simple random sample of 50 accounts was drawn. Let y denote the amount in a deposit account. The following data were obtained from the sample: 50 50 i 1 i 1 yi = 400, 000 and yi 2 = 3, 650, 000, 000 (i) Give an unbiased estimate of the average amount in a deposit account. (ii) Give 95% confidence limits for the total amount in deposit accounts. (iii) If the true total amount in deposit accounts is 110, 000, 000, comment on the validity of estimation. Q3. [4] The Orange Cab Company has just received a shipment of 5 new Checkers to be used as cabs. They wish to use these as a sample of Checkers in order to estimate the average miles per gallon of Checkers used as taxis. After one month of use the miles traveled (y) and the gallons of fuel used (x) were recorded. These are given in the table below. Use this information to estimate the average miles per gallon of Checkers and give the standard error. Miles (y) Gallons (x) 2000 400 1500 280 1000 180 1700 300 1300 245 Q4. [3+2] Consider a random sample of size 4 drawn from a population of size N by simple random sampling with replacement. Let V represent the number of distinct units in the sample. (a) Find Pr(V=v) for v = 1, 2, 4 [Hint: Pr(V=3) = (b) Find E(V) and E(1/V) 6( N 1)( N 2) N3 Q4. [4] The PCA (Private College Association) contains 400 private colleges. A simple random sample of 50 PCA colleges was selected. Data for the number of students (y) and number of teachers (x) in the samples are summarized below. Use this information to estimate the student / teacher ratio for the PCA and find the standard error of your estimate. 50 50 yi = 80, 000 50 xi = 9, 000 i 1 xi yi i 1 50 = 15, 380, 000 i 1 50 yi 2 = 140, 000, 000 xi 2 i 1 = 1, 742, 500 i 1 Q5. [4+4] Suppose that there are 1000 colleges and universities in the United States. A random sample of 60 institutions produced the following results, where y represents the number of students. Public Private 2 ni y y 25 35 100, 000 70, 000 496, 000, 000 152, 618, 000 (a) Estimate the number of the students attending private and public institutions, respectively, and give the standard error of your estimates. (b) Suppose you know that of the 1000 colleges and universities in the United States, 700 are private and 300 are public. How would you respond to the instructions in (a) for this case