Example:Using a 20-year sample, the average return of a mutual fund has been 10.5 % per year with a standard deviation of 18 %. Using these two point estimates, what is the 95% confidence interval for the return next year?
10.5 - 1.96 * 18 or -24.78 to 10.5 + 1.96 * 18 = 45.78.
In symbols, P(-24.78 < return < 45.78) = 95%.
n: Define the standard normal distribution and explain how to standardize a random variable.
The standard normal distributionis a normal distribution that has been "normalized" so that it has a mean of zero and a standard deviation of one. To standardize an observation from any given normal curve, you must calculate the observation's Z-value. The Z-value tells you how far away the given observation is from the population mean in units of standard deviation. This is how we standardize a random variable.
Z = (observation - population mean) / standard deviation = (X - µ) / σ
Example:The EPS figures for a large group of firms are normally distributed with a mean of $6 and a standard deviation of $2. What is the Z-value given and EPS of $8, that is X = 8? How about X = 2?
If X = 8, then Z = (( X - µ) / σ = (8 - 6) / 2 = +1
If X = 2, then Z = ( X - µ) / σ = (2 - 6) / 2 = -2