The Sampling Distribution of the Sample Mean
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Transcript of The Sampling Distribution of the Sample Mean
The Sampling Distribution of the The Sampling Distribution of the Sample Mean Sample Mean
The Sampling Distribution of the The Sampling Distribution of the Sample Proportion Sample Proportion
Chapter 6 Sampling Distributions(样本分布 )
Chapter 6 Sampling Distributions
Sample Mean
Partner Hours
Dunn 22
Hardy 26
Kiers 30
Malory 26
Tillman 22
Example: The law firm of Hoya and Associates has five partners. At their weekly partners meeting each reported the number of hours they billed clients for their services last week.
If two partners are selected randomly, how many different samples are possible?
Partners Total Mean 1,2 48 24 1,3 52 26 1,4 48 24 1,5 44 22 2,3 56 28 2,4 52 26 2,5 48 24 3,4 56 28 3,5 52 26 4,5 48 24
A total of 10 different samples
23/4/24 Chapter 6 Sampling Distributions
Sample Mean Frequency Relative Frequency probability
22 1 1/10
24 4 4/10
26 3 3/10
28 2 2/10
As a sampling distribution
Different samples of the same size from the same population
will yield different sample means
Chapter 6 Sampling Distributions
Sample Mean Let there be a population of units of size Let there be a population of units of size NN Consider all its samples of a fixed size Consider all its samples of a fixed size n n ((n<Nn<N)) For all possible samples of size For all possible samples of size nn, we obtain a population , we obtain a population
of sample means. That is, of sample means. That is, is a random variable which is a random variable which may have all these means as its valuesmay have all these means as its values
Before we draw the sample, the sample mean Before we draw the sample, the sample mean is a random variable.
We consider the probability distribution of the random We consider the probability distribution of the random variable variable , i.e., the probability distribution for the , i.e., the probability distribution for the population of sample meanspopulation of sample means
x
x
x
Chapter 6 Sampling Distributions
Section 6.1 The Sampling Distribution of Section 6.1 The Sampling Distribution of the Sample Meanthe Sample Mean
The The sampling distribution of the samplesampling distribution of the sample meanmean is the probability distribution of the is the probability distribution of the population of the sample means obtainable population of the sample means obtainable from all possible samples of size from all possible samples of size n n from a from a population of size population of size N.N.
Chapter 6 Sampling Distributions
Example 6.1Example 6.1 The Stock Return Case We have a population of the percent returns from six stocks
In order, the values of % return are:10%, 20%, 30%, 40%, 50%, and 60%
Label each stock A, B, C, …, F in order of increasing % return
The mean rate of return is 35% with a standard deviation of 17.078%
Any one stock of these stocks is as likely to be picked as any other of the six Uniform distribution with N = 6 Each stock has a probability of being picked of 1/6
Chapter 6 Sampling Distributions
Stock
% Return
Frequency
Relative Frequency
Stock A 10 1 1/6 Stock B 20 1 1/6 Stock C 30 1 1/6 Stock D 40 1 1/6 Stock E 50 1 1/6 Stock F 60 1 1/6 Total 6 1
The Stock Return Case 2 ﹟
Chapter 6 Sampling Distributions
Now, select all possible samples of size Now, select all possible samples of size nn = = 2 from this population of stocks of size 2 from this population of stocks of size N N = = 66 That is, select all possible pairs of stocksThat is, select all possible pairs of stocks
How to select?How to select? Sample randomlySample randomly Sample without replacementSample without replacement Sample without regard to orderSample without regard to order
The Stock Return Case 3 ﹟
Chapter 6 Sampling Distributions
Result: There are 15 possible Result: There are 15 possible samples of size samples of size nn = 2 = 2
Calculate the sample mean of each and Calculate the sample mean of each and every sampleevery sample
For example, if we choose the two stocks For example, if we choose the two stocks with returns 10% and 20%, then the sample with returns 10% and 20%, then the sample mean is 15%mean is 15%
26C
The Stock Return Case 4 ﹟
Chapter 6 Sampling Distributions
SampleMean Frequency
RelativeFrequency
15 1 1/1520 1 1/1525 2 2/1530 2 2/1535 3 3/1540 2 2/1545 2 2/1550 1 1/1555 1 1/15
The Stock Return Case 5 ﹟
Chapter 6 Sampling Distributions
The population of N = 6 stock returns has a uniform distribution.
But the histogram of n = 15 sample mean returns:
1. Seems to be centered over the same mean return of 35%, and
2. Appears to be bell-shaped and less spread out than the histogram of individual returns
Observations
Chapter 6 Sampling Distributions
Example 6.2Example 6.2 Sampling all the stocks Consider the population of returns of all 1,815
stocks listed on NYSE for 1987 See Figure 6.2(a) on next slide The mean rate of return was –3.5% with a
standard deviation of 26% Draw all possible random samples of size n = 5
and calculate the sample mean return of each Sample with a computer See Figure 6.2(b) on next slide
Chapter 6 Sampling Distributions
Chapter 6 Sampling Distributions
Observations Both histograms appear to be bell-shaped and
centered over the same mean of –3.5% The histogram of the sample mean returns looks
less spread out than that of the individual returns Statistics
Mean of all sample means: = = -3.5% Standard deviation of all possible means:
%63.115
26
nx
x
Chapter 6 Sampling Distributions
If the population of individual items is normal, then the population of all sample means is also normal
Even if the population of individual items is not normal, there are circumstances that the population of all sample means is normal (see Central Limit Theorem(中心极限定理 ) later)
General Conclusions
Chapter 6 Sampling Distributions
The mean of all possible sample means equals the The mean of all possible sample means equals the population meanpopulation mean That is,That is, =
The standard deviation sx of all sample means is less than the standard deviation of the population That is, <
Each sample mean averages out the high and the low measurements, and so are closer to m than many of the individual population measurements
General Conclusions
x
x
Chapter 6 Sampling Distributions
The empirical rule holdsThe empirical rule holds for the sampling distribution of for the sampling distribution of the sample meanthe sample mean 68.26%68.26% of all possible sample means are within (plus or of all possible sample means are within (plus or
minus) one standard deviationminus) one standard deviation ofof 95.44% of all possible observed values of x are within
(plus or minus) two ofof In the example., 95.44% of all possible sample
mean returns are in the interval [-3.5 ± (211.63)] = [-3.5 ± 23.26]
That is, 95.44% of all possible sample means are between -26.76% and 19.76%
99.73% of all possible observed values of x are within (plus or minus) three of
x
x
x
Chapter 6 Sampling Distributions
Properties of the SamplingDistribution of the Sample Mean #1
• If the population being sampled is normal, If the population being sampled is normal, then so is the sampling distribution of the then so is the sampling distribution of the sample meansample mean,
• The mean of the sampling distribution of is
=
That is, the mean of all possible sample means is the same as the population mean
x
x
x
x
Chapter 6 Sampling Distributions
• The variance The variance 22 of the sampling distribution of the sampling distribution
of of is is
That is, the variance of the sampling That is, the variance of the sampling distribution of distribution of is is directly proportional to the variance of directly proportional to the variance of
the population, and the population, and inversely proportional to the sample inversely proportional to the sample
sizesize
Properties of the SamplingDistribution of the Sample Mean #2
nx
22
x
x
x
Chapter 6 Sampling Distributions
Properties of the SamplingDistribution of the Sample Mean #3
• The standard deviation of the sampling distribution of is
That is, the standard deviation of the sampling distribution of is directly proportional to the standard
deviation of the population, and inversely proportional to the square root
of the sample size
nx
xx
x
Chapter 6 Sampling Distributions
is the point estimate of , and the larger the sample size n, the more accurate the estimate, because when n increases, decreases, is more clustered to the is more clustered to the populationpopulation
In order to reduceIn order to reduce , take bigger samples!
Notesx
xx
x
Chapter 6 Sampling Distributions
Example 6.3Example 6.3 Car Mileage Case Population of all midsize cars of a particular make and
model Population is normal with mean and standard
deviation Draw all possible samples of size Draw all possible samples of size nn Then the sampling distribution of the sample mean is Then the sampling distribution of the sample mean is
normal with meannormal with mean == and standard deviation In particular, draw samples of size:
n = 5 n = 49
nx x
Chapter 6 Sampling Distributions
2.23615
x
749
x
So, all possible sample means for n=49 will be more closely clustered around than the case of n =5
Chapter 6 Sampling Distributions
Central Limit Theorem(中心极限定理 ) #1
If the population is non-normal, what is the shape of If the population is non-normal, what is the shape of the sampling distribution of the sample means?the sampling distribution of the sample means?
In fact the sampling distribution is approximately In fact the sampling distribution is approximately normal if the sample is large enough, even if the normal if the sample is large enough, even if the population is non-normalpopulation is non-normal
by the “Central Limit Theorem”by the “Central Limit Theorem”
Chapter 6 Sampling Distributions
No matter what is the probability distribution that describes the population, if the sample size n is large enough, then the population of all possible sample means is approximately normal with mean and standard deviation
Further, the larger the sample size n, the closer the sampling distribution of the sample means is to being normal In other words, the larger n, the better the
approximation
xnx
Chapter 6 Sampling Distributions
Random Sample (x1, x2, …, xn)
Population Distribution
(, )(right-skewed)
X
as n large
n, xx
Sampling Distribution of Sample Means
(nearly normal)
x
Chapter 6 Sampling Distributions
Example 6.4Example 6.4 Effect of the Sample SizeEffect of the Sample SizeThe larger the sample The larger the sample size, the more nearly size, the more nearly normally distributed is normally distributed is the population of all the population of all possible sample meanspossible sample means
Also, as the sample size Also, as the sample size increases, the spread of increases, the spread of the sampling distribution the sampling distribution decreasesdecreases
Chapter 6 Sampling Distributions
How Large?How Large? How large is “large enough?”How large is “large enough?” If the sample size n is at least 30, then for most sampled If the sample size n is at least 30, then for most sampled
populations, the sampling distribution of sample means is populations, the sampling distribution of sample means is approximately normalapproximately normal
Refer to Figure 6.6 on next slideRefer to Figure 6.6 on next slide Shown in Fig 6.6(a) is an exponential (right Shown in Fig 6.6(a) is an exponential (right
skewed) distributionskewed) distribution In Figure 6.6(b), 1,000 samples of size In Figure 6.6(b), 1,000 samples of size nn = 5 = 5
Slightly skewed to rightSlightly skewed to right In Figure 6.6(c), 1,000 samples with In Figure 6.6(c), 1,000 samples with nn = 30 = 30
Approximately bell-shaped and normalApproximately bell-shaped and normal If the population is normal, the sampling distribution of If the population is normal, the sampling distribution of
is normal regardless of the sample sizeis normal regardless of the sample sizex
x
Chapter 6 Sampling Distributions
Example: Central Limit Theorem Simulation
Chapter 6 Sampling Distributions
Recall from Chapter 2 mileage example, = 31.5531 mpg for a sample of size n=49 With With s = 0.7992s = 0.7992
Does this give statistical evidence that the population Does this give statistical evidence that the population meanmean is greater than 31 mpg?is greater than 31 mpg? That is, does the sample mean give evidence thatThat is, does the sample mean give evidence that is is
at least at least 31 mpg31 mpg?? Calculate the probability of observing a sample mean
that is greater than or equal to 31.5531 mpg if = = 31 31 mpgmpg WantWant P(> 31.5531 if = 31)
Reasoning from Sample Distribution
x
x
Chapter 6 Sampling Distributions
Use Use ss as the point estimate for as the point estimate for so that so that
0.114349
79920
.n
x
ThenThen
84.41143.0
315531.31
5531.3131 if 5531.31
zP
zP
zPxPx
x
But But z = 4.84z = 4.84 is off the standard normal table is off the standard normal table The largest The largest zz value in the table is value in the table is 3.093.09, which has a , which has a
right hand tail area of right hand tail area of 0.0010.001
Chapter 6 Sampling Distributions
Probability that > 31.5531 when = 31= 31x
Chapter 6 Sampling Distributions
z = 4.84 > 3.09z = 4.84 > 3.09, so , so P(z ≥ 4.84) < 0.001P(z ≥ 4.84) < 0.001 That is, ifThat is, if = 31 mpg, then fewer than 1 in 1,000 of all
possible samples have a mean at least as large as samples have a mean at least as large as observedobserved
Have either of the following explanations:Have either of the following explanations: If is actually is actually 3131 mpg, then picking this sample is an mpg, then picking this sample is an
almost unbelievable thingalmost unbelievable thingOROR is not 31 mpgis not 31 mpg
Difficult to believe such a small chance would occur, so conclude that there is strong evidence thatevidence that does not equal 31 mpg.. is in fact larger than 31 mpgis in fact larger than 31 mpg
Chapter 6 Sampling Distributions
Section 6.2 The Sampling Distribution Section 6.2 The Sampling Distribution of the Sample Proportion(of the Sample Proportion( 样本比例样本比例 ))
For a population of units, we select samples of size n, and For a population of units, we select samples of size n, and calculate its proportion for the units of the sample to be calculate its proportion for the units of the sample to be fall into a particular category. fall into a particular category. is a random variable and has its probability distribution.is a random variable and has its probability distribution.
The probability distribution of all possible sample The probability distribution of all possible sample proportions is called the proportions is called the sampling distribution of the sampling distribution of the sample proportionsample proportion
p̂
p̂
Chapter 6 Sampling Distributions
the sampling distribution of isp̂
approximately normal, if n is large (meet the conditions that np≥5 and n(1-p)≥5)
has mean pp ˆ
has standard deviation
npp
p̂
1
where p is the population proportion for the category
Chapter 6 Sampling Distributions
Example 6.5Example 6.5 The Cheese Spread CaseThe Cheese Spread Case A food processing company developed a new cheese
spread spout which may save production cost. If only less than 10% of current purchasers do not accept the design, the company would adopt and use the new spout.
1000 current purchasers are randomly selected and inquired, and 63 of them say they would stop buying the cheese spread if the new spout were used. So, the sample proportion =0.063.
To evaluate the strength of this evidence, we ask: if p=0.1, what is the probability of observing a sample of size 1000 with sample proportion 0.063?≦
p̂
p̂
Chapter 6 Sampling Distributions
If p=0.10, since n=1000, np≥5 and n(1-p)≥5, is approximately normal with p̂
,1.0ˆ pp ,094868.01
ˆ
n
ppp
.001.090.3094868.0
10.0063.0
063.01.0 if 063.0
zP
zP
zPppPp
p
Chapter 6 Sampling Distributions
So, if p=0.1, the chance of observing at most 63 out of 1000 randomly selected customers do not accept the new design is less than 0.001
But such observation does occur. This means that we have extremely strong evidence that p≠0.1, and p is in fact less than 0.1
Therefore, the company can adopt the new design