- the distribution of sample means over repeated sampling from one
population
Goal:
-estimate a population mean given sample mean(s)
-as increase sample size (or the number of samples), approximate a normal curve
2. Central Limit Theorem
**Central Limit Theorem
- Given a population with mean
and variance
2
, the sampling distribution of
the mean will have a mean equal to
and a variance equal to
2
/ N.
The distribution will approach the normal
distribution as N, the sample size, increases.
-the rate at which the sampling distribution of the mean approaches normal is a function of the shape of
the parent population:
a)
-if the population is normal, the sampling distribution of the mean will be normal regardless of N
b)
-if the population is symmetric but non-normal, the sampling distribution of the mean will be
nearly normal even for quite small sample sizes
c)
-if the population is markedly skewed, sample sizes of 30 or more are required before the means
approximate a normal distribution
Examples: Sampling Distributions
1.
Use your z-tables to find the critical z-values:
a.
= .05
2-tails
b.
= .05
1-tail
c.
= .01
2-tails
d.
= .01
1-tail
2.
Use your t-tables to find the critical t-values:
a.
n = 25
= .05
2-tails
b.
n = 25
= .01
1-tail
c.
n = 30
= .01
2-tails
d.
n = 30
= .05
1-tail

3.
The national average for the verbal section of the Graduate Record Exam (GRE) is 500 with a standard
deviation of 100. A researcher uses a sampling distribution made up of samples of 100.
a.
According to the Central Limit Theorem, what is the mean of the sampling distribution of means?
b.
According to the Central Limit Theorem, what is the standard error of the mean?

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PSY1110: Modules 9 & 10 (Combined) Course Notes
Page 5
3b.
10
Hypothesis Testing (steps 4 to 8):
1)
Z-Score Hypothesis Testing with Single X-Value
z-score formula
X
***This formula is used when both
and
are known
2)
Z-Score Hypothesis Testing with Sample of X-Values
z-test formula
z
X
x
, where
x
Examples (for the z-test):
A.
Overall, Honda Accords get 28 mpg (miles per gallon) (μ = 28) on the highway with a standard
deviation of 1.3 miles (σ = 1.3). Someone believes a gas additive will
improve the mileage? After adding the
additive, a car gets 31 mpg. Does the additive improve mileage?
Set alpha equal to 0.05.
z
n