STAT371 DISCUSSION 5 October 6, 2002
1. The sampling distribution of ¯
Y
•Mean. The mean of the sampling distribution of ¯
Yis equal to the
population mean. In symbols,
µ¯
Y=µ
•Standard deviation. The standard deviation of the sampling dis-
tribution of ¯
Yis equal to the population standard deviation di-
vided by the square root of the sample size. In symbols,
σ¯
Y=σ
√n
Comparing: the sample standard deviation is
s=sP(yi−¯y)2
n−1
•Shape. (a) If the population distribution of Yis normal, then the
sampling distribution of ¯
Yis normal, regardless of the sample size
n.
(b)Central Limit Theorem If nis large, then the sampling dis-
tribution of ¯
Yis approximately normal, even if the population
distribution of Yis not normal.
2. Normal approximation of binomial distribution
•If nis large, then the binomial distribution can be approximated
by a normal distribution with
Mean = np Standard deviation = qnp(1 −p)
•If nis large, then the sampling distribution of ˆpcan be approxi-
mated by a normal distribution with
Mean = pStandard deviation = sp(1 −p)
n
•The normal approximation to the binomial distribution is fairly
good if both np and n(1 −p) are at least equal to 5.
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