Assuming there are only 30 stocks in the world, and the annual rates of return (Xi) are following:
Xi Xi+100 Xi*100 Frequency(Number of stocks)
0.01 100.01 1 1
0.03 100.03 3 2
0.05 100.05 5 7
0.07 100.07 7 3
0.09 100.09 9 2
0.11 100.11 11 4
0.13 100.13 13 9
0.15 100.15 15 2

a) What are the mean and median rates of return?

b) What are the variance and standard deviation of the rates of return?

c) Prove the following properties of the mean using the data.
E(Xi+100) = 100 + E(Xi)
E(Xi*100) = 100* E(Xi)

d) Prove the following properties of the variance using the data.
Var(Xi*100) = 10000* Var(Xi)
Var(Xi+100) = Var(Xi)

e) Suppose above data is a sample. Find the sample mean and sample stand deviation.

f) Explain the sample the sample mean distributions?

g) Find the confidence intervals of the population mean at 95% and 99%.