图样本平均数的抽样分配

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Chapter 7
Sampling Distributions
Basic Business Statistics
10th Edition
Learning Objectives
In this chapter, you learn:
The concept of the sampling distribution
To compute probabilities related to the sample mean and the sample proportion
The importance of the Central Limit Theorem
To distinguish between different survey sampling methods
To evalua te survey worthiness and survey errors
Sampling Distributions
Sampling Distributions
Sampling Distribution of the Mean
Sampling Distribution of the Proportion
Sampling Distributions
A sampling distribution is a distribution of all of the possible values of a statistic for a given size sample selected from a population
Developing a
Sampling Distribution
Assume there is a population …
Population size N=4
Random variable, X,
is age of individuals
Values of X: 18, 20,
22, 24 (years)
A
B
C
D
Developing a
Sampling Distribution
.3
.2
.1
0
18 20 22 24
A B C D
Uniform Distribution
P(x)
x
(continued)
Summary Measures for the Population Distribution:
Now consider all possible samples of size n=2
16 possible samples (sampling with replacement)
(continued)
Developing a
Sampling Distribution
16 Sample Means
24,24
24,22
24,20
24,18
24
22,24
22,22
22,20
22,18
22
20,24
20,22
20,20
20,18
20
18,24
18,22
18,20
18,18
18
24
22
20
18
2nd Observation
1st
Obs
Sampling Distribution of All Sample Means
18 19 20 21 22 23 24
0
.1
.2
.3
P(X)
X
Sample Means Distribution
16 Sample Means
_
Developing a
Sampling Distribution
(continued)
(no longer uniform)
_
Summary Measures of this Sampling Distribution:
Developing a
Sampling Distribution
(continued)
Comparing the Population with its Sampling Distribution
18 19 20 21 22 23 24
0
.1
.2
.3
P(X)
X
18 20 22 24
A B C D
0
.1
.2
.3
Population
N = 4
P(X)
X
_
Sample Means Distribution
n = 2
_
Sampling Distribution
of the Mean
Sampling Distributions
Sampling Distribution of the Mean
Sampling Distribution of the Proportion
Standard Error of the Mean
Different samples of the same size from the same population will yield different sample means
A measure of the variability in the mean from sample to sample is given by the Standard Error of the Mean:
(This assumes that sampling is with replacement or
sampling is without replacement from an infinite population)
Note that the standard error of the mean decreases as the sample size increases
If the Population is Normal
If a population is normal with mean μ and standard deviation σ, the sampling distribution of is also normally distributed with
and

Z-value for Sampling Distribution
of the Mean
Z-value for the sampling distribution of :
where: = sample mean
= population mean
= population standard deviation
n = sample size
Sampling Distribution Properties

(i.e. is unbiased )
Normal Population Distribution
Normal Sampling Distribution
(has the same mean)
Sampling Distribution Properties
As n increases,
decreases
Larger sample size
Smaller sample size
(continued)
If the Population is not Normal
We can apply the Central Limit Theorem:
Even if the population is not normal,
…sample means from the population will be approximately normal as long as the sample size is large enough.
Properties of the sampling distribution:
and
Central Limit Theorem
n↑
As the sample size gets large enough…
the sampling distribution becomes almost normal regardless of shape of population
If the Population is not Normal
Population Distribution
Sampling Distribution
(becomes normal as n increases)
Central Tendency
Variation
Larger sample size
Smaller sample size
(continued)
Sampling distribution properties:
How Large is Large Enough
For most distributions, n > 30 will give a sampling distribution that is nearly normal
For fairly symmetric distributions, n > 15
For normal population distributions, the sampling distribution of the mean is always normally distributed
Example
Suppose a population has mean μ = 8 and standard deviation σ = 3. Suppose a random sample of size n = 36 is selected.
What is the probability that the sample mean is between 7.8 and 8.2
Example
Solution:
Even if the population is not normally distributed, the central limit theorem can be used (n > 30)
… so the sampling distribution of is approximately normal
… with mean = 8
…and standard deviation
(continued)
Example
Solution (continued):
(continued)
Z
7.8 8.2
-0.4 0.4
Sampling Distribution
Standard Normal Distribution
.1554 +.1554
Population Distribution












Sample
Standardize
X
Distribution of Sample Means
for Various Sample Sizes
Exponential
Population
n = 2
n = 5
n = 30
Uniform
Population
n = 2
n = 5
n = 30
Distribution of Sample Means
for Various Sample Sizes
U Shaped
Population
n = 2
n = 5
n = 30
Normal
Population
n = 2
n = 5
n = 30
图—样本平均数的抽样分配
应用中央极限定理应注意事项
表—样本平均数的抽样分配
Sampling Distribution
of the Proportion
Sampling Distributions
Sampling Distribution of the Mean
Sampling Distribution of the Proportion
Population Proportions
π = the proportion of the population having
some characteristic
Sample proportion ( p ) provides an estimate
of π:
0 ≤ p ≤ 1
p has a binomial distribution
(assuming sampling with replacement from a finite population or without replacement from an infinite population)
Sampling Distribution of p
Approximated by a
normal distribution if:

where
and
(where π = population proportion)
Sampling Distribution
P( ps)
.3
.2
.1
0
0 . 2 .4 .6 8 1
p
Z-Value for Proportions
Standardize p to a Z value with the formula:
Example
If the true proportion of voters who support Proposition A is π = 0.4, what is the probability that a sample of size 200 yields a sample proportion between 0.40 and 0.45
i.e.: if π = 0.4 and n = 200, what is
P(0.40 ≤ p ≤ 0.45)
Example
if π = 0.4 and n = 200, what is
P(0.40 ≤ p ≤ 0.45)
(continued)
Find :
Convert to standard normal:
Example
if π = 0.4 and n = 200, what is
P(0.40 ≤ p ≤ 0.45)
Z
0.45
1.44
0.4251
Standardize
Sampling Distribution
Standardized
Normal Distribution
(continued)
Use standard normal table: P(0 ≤ Z ≤ 1.44) = 0.4251
0.40
0
p
Reasons for Drawing a Sample
Less time consuming than a census
Less costly to administer than a census
Less cumbersome and more practical to administer than a census of the targeted population
Types of Samples Used
Nonprobability Sample
Items included are chosen without regard to their probability of occurrence
Probability Sample
Items in the sample are chosen on the basis of known probabilities
Types of Samples Used
Quota
Samples
Non-Probability Samples
Judgement
Chunk
Probability Samples
Simple
Random
Systematic
Stratified
Cluster
Convenience
(continued)
Probability Sampling
Items in the sample are chosen based on known probabilities
Probability Samples
Simple
Random
Systematic
Stratified
Cluster
Simple Random Samples
Every individual or item from the frame has an equal chance of being selected
Selection may be with replacement or without replacement
Samples obtained from table of random numbers or computer random number generators
Systematic Samples
Decide on sample size: n
Divide frame of N individuals into groups of k individuals: k=N/n
Randomly select one individual from the 1st group
Select every kth individual thereafter
N = 64
n = 8
k = 8
First Group
Stratified Samples
Divide population into two or more subgroups (called strata) according to some common characteristic
A simple random sample is selected from each subgroup, with sample sizes proportional to strata sizes
Samples from subgroups are combined into one
Population
Divided
into 4
strata
Sample
Cluster Samples
Population is divided into several "clusters," each representative of the population
A simple random sample of clusters is selected
All items in the selected clusters can be used, or items can be chosen from a cluster using another probability sampling technique
Population divided into 16 clusters.
Randomly selected clusters for sample
Advantages and Disadvantages
Simple random sample and systematic sample
Simple to use
May not be a good representation of the population's underlying characteristics
Stratified sample
Ensures representation of individuals across the entire population
Cluster sample
More cost effective
Less efficient (need larger sample to acquire the same level of precision)
Evaluating Survey Worthiness
What is the purpose of the survey
Is the survey based on a probability sample
Coverage error – appropriate frame
Nonresponse error – follow up
Measurement error – good questions elicit good responses
Sampling error – always exists
Types of Survey Errors
Coverage error or selection bias
Exists if some groups are excluded from the frame and have no chance of being selected
Nonresponse error or bias
People who do not respond may be different from those who do respond
Sampling error
Variation from sample to sample will always exist
Measurement error
Due to weaknesses in question design, respondent error, and interviewer's effects on the respondent
Types of Survey Errors
Coverage error
Non response error
Sampling error
Measurement error
Excluded from frame
Follow up on nonresponses
Random differences from sample to sample
Bad or leading question
(continued)
Chapter Summary
Introduced sampling distributions
Described the sampling distribution of the mean
For normal populations
Using the Central Limit Theorem
Described the sampling distribution of a proportion
Calculated probabilities using sampling distributions
Described different types of samples and sampling techniques
Examined survey worthiness and types of survey errors
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