Suppose that y1 y 2 yn is a random sample of size n from

Suppose that y₁, y ₂, . . . , y_n is a random sample
of size n from a normal distribution where
is known. Depending on how the tail-area probabilities
are split up, an infinite number of random intervals
having a 95% probability of containing can be cons
tructed. What is unique about the particular interval
(y 1.96 · /n , y +1.96 · /n)?

Solution

The z value is calculated by using the formula

Z = (x - mean)/sigma

The value of Z equal to 1.96 indicates the 95% probability of containing the sample, hence the range of

[y - 1.96 * (sigma)^(1/2)/n] to [y + 1.96 * (sigma)^(1/2)/n]

Hence the value of 1.96 is a special value which corresponds to 95% area under the curve

P(-1.96<z<1.96) = P(Z<1.96) - P(Z<-1.96) = 0.95004 - 0.49996 = 0.90008