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SAMPLING DISTRIBUTIONS OF TEST STATISTICS
FROM NORMALLY DISTRIBUTED POPULATIONS
Case I. The random variable X is normally distributed with mean, µ,
and variance, σ2.
1. If the z-score is computed for each value of x in the
population, then the random variable Z has a standard normal
distribution.
2. If each value of z is squared, then Z2 has a Chi-Square
distribution with one degree of freedom. This distribution is
denoted by χ 21.
Case II: The population from which the samples are drawn is normal
with mean, µ, and standard deviation, σ. Each sample has n
observations.
1. The sampling distribution of
, the sample mean, is
normal with mean, µ, and standard deviation
2. The sampling distribution of
the z-score of the
sample mean, is standard normal.
3.
is the sample variance.
has a Chi-Squared distribution with n - 1 degrees of freedom.
This distribution is denoted by χ 2n-1.
5. The random variable T is defined as,
n-1 degrees of freedom. Hence, it can be shown that
has a t-distribution.
It has
Case III: Independent samples are drawn from two different
populations. Each population is normal. Population 1 has mean, µ1,
and standard deviation, σ1; population 2 has mean, µ2, and standard
deviation, σ2.
1. Let Y = X1 + X2. The sampling distribution of Y is normal
with mean, µ1 + µ2, and variance, σ21 + σ22.
2. Let Y = a1X1 +a2X2. The sampling distribution of Y is normal
with mean, a1µ1 + a2µ2, and variance, a21σ21 + a22σ22.
3. A sample with n1 observations is chosen from population 1,
and a sample with n2 observations is chosen from population 2.
Let K =
-
. The sampling distribution of K is normal
with mean, µ1 - µ2, and variance,
4. The z-score of K,
is
standard normal.
5. Let Z1 and Z2 be the z-scores of
then W = Z21 + Z22 is χ 22.
6. If U is
X1 and X2,
χ2r and V is χ2s, then U + V is χ2r+s and
is
Fr,s.
7.A sample with n1 observations is chosen from population 1,
and a sample with n2 observations is chosen from population 2.
=
If
σ1
=
σ2,
then
χ2df with df = n1-1+n2-1 = n1+n2-2.
the
sampling
distribution
of
is t with n1 + n2 - 2 degrees
of freedom.
Exercises
µ1 = 50 and σ12 = 9. µ2 = 60 and σ22 = 16.
1) If a sample of 81 observations is chosen from population 1, what
is the probability that
will be greater than 51.3?
2) If a sample of 64 observations is chosen from population 2, what
is the probability that
will be greater than 51.3?
3) If a sample of 81 observations is chosen from population 1
and a sample of 64 observations is chosen from population 2, what
is the probability that
-
will be greater than 0?
4) If a sample of 15 observations is selected from sample 1, what
is the probability that s12 will be greater than 32.76? What is the
probability that s12 will be greater than 7?