STA220H5 Chapter Notes - Chapter 6.3: Central Limit Theorem, Standard Deviation, Sampling Distribution
Theorem 6.1
If a random sample of n observations is selected from a population with a normal
distribution, the sampling distribution of x will be a normal distribution.
Theorem 6.2: central limit theorem
Consider a random sample of n observations selected from a population (any
population) with mean u and standard deviation a. then, when n is sufficiently large,
the sampling distribution of x will be approximately a normal distribution with
mean ux=u and standard deviation ax=a/n. the larger the sample size, the better will
be the normal approximation to the sampling distribution of x.*
Sampling distribution of p
1. Mean of the sampling distribution is equal to the true binomial proportion, p;
that is, E (p)=p. consequently, p is an unbiased estimator of p.
2. Standard deviation of the sampling distribution is equal to p (1-p)/n; that is,
ap=p (1-p)/n.
3. For large samples, the sampling distribution is approximately normal. (a
sample is considered large is np>15 and n(1-p)>15.)
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