7.3    Distribution of the Sample Mean; Central Limit Theorem

We'll use two important results about normal random variables.

1. If X is a normal r. v. with mean m, variance s², and c is a constant, then Y = cX is normal, with mean cm, variance c²s².
2. If X₁ and X₂ are independent normal random variables with means m₁ and m₂  and variances s₁² and s₂², then  X₁+X₂  is normally distributed,with mean m₁+m₂, variance s₁²+s₂².

To summarize:

• a constant times a normal r.v. is normal
• the sum of normals is normal.

if X &  Y have the same moment generating function, m_X(t) = m_Y(t),
then X & Y have the same distribution (density function).

Distribution of the sample mean
 
From the above two properties it follows that:

if independent random variables X₁, X₂, ..., X_n are a random sample from a normal distribution with mean m, variance s², then the sample mean  is normally distributed with mean  m = m  and
variance  s²  =  s²/n .

Using the information about the distribution of 

ex:

Raising trout in a fish hatchery; suppose lengths (of 2 year olds) are normally distributed, with mean m = 7.3 inches, standard deviation s = 1.6 inches.

Take samples of size 70, and compute the sample mean  for each sample; then the sample means  will be normally distributed, with mean  m  = 7.3 inches and standard deviation  s  = 1.6/sqrt(70)  =  .19 inches.

Graphs of densities:  the density graph for the sample means from the various samples will be a normal curve, centered at the same location (7.3 inches) as the population density curve, but will be much narrower: the standard deviation of the sample means is just .19, while that for the population is 1.6. Thus the values of the sample mean for various samples will be centered at the population mean, but will tend to vary much less on either side than individuals from the population.

By the normal probability rule, since  is normally distributed,

the probability that  will lie within 2 standard deviations of its mean is .95
i.e.,
the probability that  will lie within .38 units of the population mean is .95
(since the standard deviation of  is .19, and the mean of  is the same as the population mean)
i.e.,
95% of samples we choose will have  lying within .38 units of the population mean
Thus:
For 95% of the samples we draw,  will lie within .38" of the value of m
This gives us an idea of how reliable the value of  from a sample will be as an estimate of the value of m: 95% of the time, the population mean m will be within .38 units of the value found for the sample mean . This is the fundamental idea behind confidence intervals (discussed in the next section).

Central Limit Theorem

Let X₁, X₂, ..., X_n be a random sample from a population having any distribution (with mean m, variance s²), not necessarily a normal distribution; then for large n, the sample mean  will be approximately normally distributed (with mean m, variance s²/n ).

• this is surprising: regardless of distribution of population, distribution of sample mean will be approximately normal for large n!!
• thus can use normal calculations with the sample mean even when distribution of population isn’t  normal
• proof involves looking at moment generating functions