Download Chapter 8 Key Points Packet File

A.P. Statistics
Chapter 8 Key facts
[8-1]
Definitions
Any quantity computed from values in a sample is called a statistic.
The observed value of a statistic depends on the particular sample selected from the
population; typically, it varies from sample to sample. This variability is called sampling
variability.The distribution of the possible values of a statistic is called its sampling
distribution.
[8-2]
General Properties of the Sampling Distribution of x
Let x denote the mean of the observations in a random sample of size n from a
population having mean μ and standard deviation σ. Denote the mean value of the x
distribution by  x and the standard deviation of the x distribution by  x . Then the
following rules hold.
Rule 1.  x  
x 

Rule 2.
This rule is exact if the population is infinite, and is approximately
n
correct if the population is finite and no more than 10% of the population is
included in the sample.
Rule 3.
When the population distribution is normal, the sampling distribution of x is
also normal for any sample size n.
Rule 4.
(Central Limit Theorem) When n is sufficiently large, the sampling
distribution of x is well approximated by a normal curve, even when the
population distribution is not itself normal.
*If n is large or the population distribution is normal, the standardized variable
z
x  x
x

x 

n
has (at least approximately) a standard normal (z) distribution.
*The Central Limit Theorem can safely be applied if n exceeds 30.
Illustration of the CLT(Central Limit Theorem)
NORMAL POPULATION
Normal distribution of platelet size x with  = 8.25 and  = .75
Sample histograms for x based on 500 samples, each consisting of n
observations:
(a) n = 5;
(b) n = 10;
(c) n = 20;
(d) n = 30
NON-NORMAL POPULATION
The population distribution (μ = 9.841)
Four histograms of 500 x values for Example 8.5
[8-3]
General Properties of the Sampling Distribution of p
Let p be the proportion of successes in a random sample of size n from a population
whose proportion of S’s is π. Denote the mean value of p by  p and the standard
deviation by  p . Then the following rules hold.
Rule 1.
p  
Rule 2.
p 
 (1   )
This rule is exact if the population is infinite, and is
n
approximately correct if the population is finite and no more than 10% of the
population is included in the sample.
Rule 3. When n is large and π is not too near 0 or 1, the sampling distribution of p is
approximately normal.
*The farther the value of π is from .5, the larger n must be for a normal approximation to
the sampling distribution of p to be accurate. A conservative rule of thumb is that if both
nπ 12 Histograms of 500 values of p (
= .07)
Illustration of the sampling distribution of p
Histograms of 500 values of p (  = .07/population positively skewed)