Let Z be a random variable with distrobution N01 Denote the

Let Z be a random variable with distrobution N(0,1). Denote the cdf of Z by the normal. Let Z₁, Z₂, ...,Z_n be a random sample from N(0,1) and Zbar be the sample mean.

A. Express the cdf of Zbar in terms of the normal and n

D. explain in words the meaning of the result form parts a and b

Solution

We have z bar = average of all z1,z2....zn

Hence Z =0 is always less than z bar

c) Thus P(Z<=c) occupies less area than P(Z bar <=c) hence inequality holds good.

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D. Z the normal std variable has less prob for less than c than the mean of z1, z2... zn