Sampling Distribution of a Normal Variable. Given a random variable X. Suppose
that the population distribution of X is known to be normal, with mean µ and.
Ismor Fischer, 5/29/2012
5.2-1
5.2 Formal Statement and Examples
Sampling Distribution of a Normal Variable Given a random variable X. Suppose that the population distribution of X is known to be normal, with mean µ and variance σ 2, that is, X ~ N(µ, σ). Then, for any sample size n, it follows that the sampling distribution of X is normal,
σ2
σ
. with mean µ and variance n , that is, X ~ Nµ, n
Comments:
σ n
is called the “standard error of the mean,” denoted SEM, or more simply, s.e.
The corresponding Z-score transformation formula is Z =
X −µ ~ N(0, 1). σ/ n
Example: Suppose that the ages X of a certain population are normally distributed, with mean µ = 27.0 years, and standard deviation σ = 12.0 years, i.e., X ~ N(27, 12). The probability that the age of a single randomly selected individual is less than 30 years
is P(X < 30) = PZ
