Question

Suppose that the monthly return of stock A is approximately normally distributed with mean µ and standard deviation σ, where µ and σ are two unknown parameters. We want to learn more about the population mean µ, so we collect the monthly returns of stock A in nine randomly selected months. The returns are given (in percentage) as follows:

0.3, 1.3, 1.5, −0.6, −0.2, 0.8, 0.8, 0.9, −1.2

Answer the following questions about the confidence intervals for µ.

(a) Construct the 90% confidence interval for µ.

(b) If the population (monthly return of stock A) is not normally distributed, can we construct the confidence interval for µ the way we do in (a)? Why or why not?

Answer 1

First we need to find sample mean ( $\bar{x}$ ) and sample standard deviation (s) of the given data set

We can find the sample mean and standard deviation of the both years using excel function =AVERAGE( ) and =STDEV.S() respectively.

1 0.3 2 1.3 3 1.5 4 0.6 AVERAGE(A1:A9)0.4 STDEV.S(A1:A9) 0.9028 5 -0.2 6 0.8 7 0.8 8 0.9

sample mean ( $\bar{x}$ ) = 0.4 and sample standard deviation (s) = 0.9028

Now we have to find margin of error E

E = $\frac{t*S}{\sqrt{n}}$ ; t is critical value follows t distribution with degrees of freedom (df) = n-1

We are given confidence level = 0.90

α = 1 - confidence level = 1 - 0.90 = 0.1

degrees of freedom (d.f) = 9 - 1 = 8

So t = 1.8595

E =

1.8595 0.9028

E = 0.5596

Lower bound = $\bar{x}$ - E = 0.4 - 0.5596 = -0.1596

Upper bound = $\bar{x}$ + E = 0.4 + 0.5596 = 0.9596

a) The 90% confidence interval for µ is ( -0.1596 , 0.9596 )

b) If the population (monthly return of stock A) is not normally distributed, then we can not construct the confidence interval for µ the way we do in (a); because the basic assumption for confidence interval is that the data must follow normal distribution.