CHAPTER 8,
Sampling Distributions
to accompany
Introduction to Business Statistics
fourth edition,by Ronald M,Weiers
Presentation by Priscilla Chaffe-Stengel
Donald N,Stengel
? 2002 The Wadsworth Group
Chapter 8 - Learning Objectives
? Determine the sampling distributions of:
– Means.
– Proportions.
? Explain the Central Limit Theorem.
? Determine the effect on the sampling
distribution when the samples are
relatively large compared to the
population from which they are drawn.
? 2002 The Wadsworth Group
Sampling Distribution of the Mean
? When the population is normally distributed
–Shape,Regardless of sample size,the
distribution of sample means will be normally
distributed.
–Center,The mean of the distribution of sample
means is the mean of the population,Sample
size does not affect the center of the distribution.
–Spread,The standard deviation of the
distribution of sample means,or the standard
error,is,
nx
ss =
? 2002 The Wadsworth Group
Standardizing a Sample Mean
on a Normal Curve
? The standardized z-score is how far above or
below the sample mean is compared to the
population mean in units of standard error.
–,How far above or below” = sample mean minus μ
–,In units of standard error” = divide by
? Standardized sample mean
n
xz
s
m=m-= –
errorstandard
mean sample
n
s
? 2002 The Wadsworth Group
Central Limit Theorem
? According to the Central Limit Theorem
(CLT),the larger the sample size,the more
normal the distribution of sample means
becomes,The CLT is central to the concept
of statistical inference because it permits us
to draw conclusions about the population
based strictly on sample data without
having knowledge about the distribution of
the underlying population.
? 2002 The Wadsworth Group
Sampling Distribution of the Mean
? When the population is not normally distributed
–Shape,When the sample size taken from
such a population is sufficiently large,the
distribution of its sample means will be
approximately normally distributed regardless
of the shape of the underlying population
those samples are taken from,According to
the Central Limit Theorem,the larger the
sample size,the more normal the distribution
of sample means becomes.
? 2002 The Wadsworth Group
Sampling Distribution of the Mean
? When the population is not normally distributed
–Center,The mean of the distribution of
sample means is the mean of the population,μ,
Sample size does not affect the center of the
distribution.
–Spread,The standard deviation of the
distribution of sample means,or the standard
error,is,
nx
ss =
? 2002 The Wadsworth Group
Example,Standardizing a Mean
? Problem 8.45,When a production machine is properly
calibrated,it requires an average of 25 seconds per unit
produced,with a standard deviation of 3 seconds,For a
simple random sample of n = 36 units,the sample mean is
found to be 26.2 seconds per unit,When the machine is
properly calibrated,what is the probability that the mean
for a simple random sample of this size will be at least 26.2
seconds?
–
– Standardized sample mean:
? 2002 The Wadsworth Group
3,25,2.26 === smx
40.2
36
3
252.26 =-=z
0 0 8 2.0)40.2()2.26( =?=? zPxP
Sampling Distribution of the
Proportion
? When the sample statistic is generated by a
count not a measurement,the proportion of
successes in a sample of n trials is p,where
–Shape,Whenever both n p and n(1 –p) are
greater than or equal to 5,the distribution of
sample proportions will be approximately
normally distributed.
? 2002 The Wadsworth Group
Sampling Distribution of the
Proportion
? When the sample proportion of successes in a
sample of n trials is p,
–Center,The center of the distribution of
sample proportions is the center of the
population,p.
–Spread,The standard deviation of the
distribution of sample proportions,or the
standard error,is sp = p× (1–p)n,
? 2002 The Wadsworth Group
Standardizing a Sample
Proportion on a Normal Curve
? The standardized z-score is how far above or
below the sample proportion is compared to the
population proportion in units of standard error.
–,How far above or below” = sample p –p
–,In units of standard error” = divide by
? Standardized sample proportion
n
pz
)–1(
–
errorstandard
proportionsample
p×p
p=p-=
np )–1( p×p=s
? 2002 The Wadsworth Group
Example,Standardizing a
Proportion
? Problem 8.40,The campaign manager for a political
candidate claims that 55% of registered voters favor the
candidate over her strongest opponent,Assuming that this
claim is true,what is the probability that in a simple
random sample of 300 voters,at least 60% would favor the
candidate over her strongest opponent?
– p = 0.55,p = 0.60,n = 300
– Standardized sample proportion:
? 2002 The Wadsworth Group
74.1
300
)55.01(55.0
55.060.0 =
-
-=z
0 4 0 9.0)74.1()60.0( =?=? zPpP
When the Population is Finite
? Finite Population Correction (FPC) Factor:
–Rule of Thumb,Use FPC when n > 5%?N.
–Apply to,Standard errors of mean and
proportion.
FPC = --N nN 1
? 2002 The Wadsworth Group