Unformatted text preview: Chapter 7
Sampling and Sampling
D istr ibutions Simple Random Sampling Point Estimation Introduction to Sampling Distributions Sampling Distribution of x Sampling Distribution of p Other Sampling Methods Statistical Inference The purpose of statistical inference is to obtain information about a population from information contained in a sample. A population is the set of all the elements of interest. A sample is a subset of the population. Statistical Inference The sample results provide only estimates of the values of the population characteristics. With proper sampling methods, the sample results can provide “good” estimates of the population characteristics. A parameter is a numerical characteristic of a population. Simple Random Sampling:
F inite Population
• Finite populations are often defined by lists such as:
– Organization membership roster
– Credit card account numbers
– Inventory product numbers
s A simple random sample of size n from a finite population of size N is a sample selected such that each possible sample of size n has the same probability of being selected. Simple Random Sampling:
F inite Population Replacing each sampled element before selecting subsequent elements is called sampling with replacement. Sampling without replacement is the procedure used most often. In large sampling projects, computergenerated random numbers are often used to automate the sample selection process. Simple Random Sampling:
I nfinite Population
• Infinite populations are often defined by an ongoing process whereby the elements of the population consist of items generated as though the process would operate indefinitely.
s A simple random sample from an infinite population is a sample selected such that the following conditions are satisfied.
• Each element selected comes from the same population.
• Each element is selected independently. Simple Random Sampling:
I nfinite Population In the case of infinite populations, it is impossible to obtain a list of all elements in the population. The random number selection procedure cannot be used for infinite populations. Point Estimation In point estimation we use the data from the sample to compute a value of a sample statistic that serves as an estimate of a population parameter. x We refer to as the point estimator of the population mean µ . s is the point estimator of the population standard deviation σ .
p is the point estimator of the population proportion p. Sampling Er r or When the expected value of a point estimator is equal to the population parameter, the point estimator is said to be unbiased. The absolute value of the difference between an unbiased point estimate and the corresponding population parameter is called the sampling error. Sampling error is the result of using a subset of the population (the sample), and not the entire population. Statistical methods can be used to make probability statements about the size of the sampling error. Sampling Error
s The sampling errors are:
 x − µ for sample mean  s −σ   p − p for sample standard deviation
for sample proportion Example: St. Andr ew ’s St. Andrew’s College receives
900 applications annually from
prospective students. The
application form contains a variety of information
including the individual’s
scholastic aptitude test (SAT) score and whether or not the individual desires oncampus housing. Example: St. Andr ew ’s The director of admissions
would like to know the
following information:
– the average SAT score for the 900 applicants, and
– the proportion of
applicants that want to live on campus. Example: St. Andr ew ’s
We will now look at two
alternatives for obtaining the
desired information.
• Conducting a census of the entire 900 applicants
• Selecting a sample of 30 applicants, using Excel Conducting a Census
s s If the relevant data for the entire 900 applicants were in the college’s database, the population parameters of interest could be calculated using the formulas presented in Chapter 3.
We will assume for the moment that conducting a census is practical in this example. Conducting a Census
• Population Mean SAT Score ∑x
µ= i 900 = 990 • Population Standard Deviation for SAT Score
σ= ( x i − µ )2
∑
900 = 80 • Population Proportion Wanting OnCampus Housing
648
p=
= .72
900 Simple Random Sampling Now suppose that the necessary data on the current year’s applicants were not yet entered in the college’s database. Furthermore, the Director of Admissions must obtain estimates of the population parameters of interest for a meeting taking place in a few hours. She decides a sample of 30 applicants will be used. The applicants were numbered, from 1 to 900, as their applications arrived. Simple Random Sampling:
U sing Excel
• Taking a Sample of 30 Applicants Step 1: Assign a random number to each of the 900 applicants. Excel’s RAND function generates random numbers between 0 and 1 Step 2: Select the 30 applicants corresponding to the 30 smallest random numbers. Point Estimation
x
• as Point Estimator of µ ∑x
x=
30 i = 29, 910
= 997
30 • s as Point Estimator of σ
s= ∑ (x i − x )2 29 = 163, 996
= 75.2
29 p
• as Point Estimator of p
p = 20 30 = .68 Note: Different random numbers would have
identified a different sample which would have
resulted in different point estimates. Summary of Point Estimates
Obtained from a Simple Random Sample
Population
Parameter µ = Population mean SAT score 990 x = Sample mean 997 σ = Population std. deviation for SAT score 80 s = Sample std. deviation for SAT score 75.2 p = Population pro portion wanting campus housing Parameter
Value Point
Estimator Point
Estimate .72 SAT score p = Sample pro portion wanting campus housing .68 x
Sampling Distribution of • Process of Statistical Inference
A simple random sample
of n elements is selected
from the population. Population with mean
µ = ? The sample data provide a value for
x
the sample mean . x
The value of is used to
make inferences about
the value of µ . x
Sampling Distribution of x
The sampling distribution of is the probability
distribution of all possible values of the sample x
mean .
Expected Value of x
x
E( ) = µ where: µ = the population mean x
Sampling Distribution of Standard Deviation of x
Finite Population Infinite Population σ N −n
σx = ( )
n N −1 σ
σx =
n • A finite population is treated as being infinite if n/N < .05.
( N − n ) / ( N − 1)
• is the finite correction factor. σx
• is referred to as the standard error of the mean. Form of the Sampling Distribution of x
If we use a large (n > 30) simple random sample, the
central limit theorem enables us to conclude that the
x
sampling distribution of can be approximated by
a normal distribution.
When the simple random sample is small (n < 30),
x
the sampling distribution of can be considered
normal only if we assume the population has a
normal distribution. x
Sampling Distribution of for SAT Scores Sampling
Distribution
x
of σx = x E( x ) = 990 σ
80
=
= 14.6
n
30 x
Sampling Distribution of for SAT Scores What is the probability that a simple random sample
of 30 applicants will provide an estimate of the
population mean SAT score that is within +/−
10 of
the actual population mean µ ? ?
x In other words, what is the probability that will be
between 980 and 1000? x
Sampling Distribution of for SAT Scores
Step 1: Calculate the zvalue at the upper endpoint of the interval.
z = (1000 − 990)/14.6= .68
Step 2: Find the area under the curve to the left of the upper endpoint.
P(z < .68) = .7517 x
Sampling Distribution of for SAT Scores
Cumulative Probabilities for the Standard Normal Distribution
z .00 .01 .02 .03 .04 .05 .06 .07 .08 .09 . . . . . . . . . . . .5 .6915 .6950 .6985 .7019 .7054 .7088 .7123 .7157 .7190 .7224 .6 .7257 .7291 .7324 .7357 .7389 .7422 .7454 .7486 .7517 .7549 .7 .7580 .7611 .7642 .7673 .7704 .7734 .7764 .7794 .7823 .7852 .8 .7881 .7910 .7939 .7967 .7995 .8023 .8051 .8078 .8106 .8133 .9 .8159 .8186 .8212 .8238 .8264 .8289 .8315 .8340 .8365 .8389 . . . . . . . . . . . x
Sampling Distribution of for SAT Scores Sampling
Distribution
x
of σ x = 14.6 Area = .7517 990 1000 x x
Sampling Distribution of for SAT Scores
Step 3: Calculate the zvalue at the lower endpoint of the interval.
z = (980 − 990)/14.6= .68
Step 4: Find the area under the curve to the left of the lower endpoint.
P(z < .68) = .2483 x
Sampling Distribution of for SAT Scores Sampling
Distribution
x
of σ x = 14.6 Area = .2483 980 990 x x
Sampling Distribution of for SAT Scores
Step 5: Calculate the area under the curve between the lower and upper endpoints of the interval.
P(.68 < z < .68) = P(z < .68) − P(z < .68)
= .7517 − .2483
= .5034
The probability that the sample mean SAT score will
be between 980 and 1000 is: x
P(980 < < 1000) = .5034 x
Sampling Distribution of for SAT Scores Sampling
Distribution
x
of σ x = 14.6 Area = .5034 980 990 1000 x Relationship Between the Sample Size
x and the Sampling Distribution of Suppose we select a simple random sample of 100 applicants instead of the 30 originally considered. E( ) = µ regardless of the sample size. In our
x
x example, E( ) remains at 990. Whenever the sample size is increased, the standard
σx error of the mean is decreased. With the increase in the sample size to n = 100, the standard error of the meanis decreased to:
σ
80
σx =
=
= 8.0
n
100 Relationship Between the Sample Size
x and the Sampling Distribution of With n = 100,
σ x = 8
With n = 30,
σ x = 14.6 E( x ) = 990 x Relationship Between the Sample Size
x and the Sampling Distribution of Recall that when n = 30, P(980 < < 1000) = .5034.
x We follow the same steps to solve for P(980 < x 1000)
< when n = 100 as we showed earlier when n = 30. Now, with n = 100, P(980 < < 1000) = .7888.
x Because the sampling distribution with n = 100 has a
x smaller standard error, the values of have less variability and tend to be closer to the population
x mean than the values of with n = 30. Relationship Between the Sample Size
x and the Sampling Distribution of Sampling
Distribution
x
of σx = 8 Area = .7888 980 990 1000 x Sampling Distribution of p
s Making Inferences about a Population Proportion
A simple random sample
of n elements is selected
from the population. Population with proportion
p = ? The sample data provide a value for the
p
sample proportion . p
The value of is used
to make inferences
about the value of p. Sampling Distribution of p
p
The sampling distribution of is the probability
distribution of all possible values of the sample
p
proportion .
Expected Value of p E ( p) = p
where: p = the population proportion Sampling Distribution of p
Standard Deviation of p
Finite Population σp = Infinite Population p (1 − p ) N − n
n
N −1 σp = p (1 − p )
n σ p is referred to as the standard error of the proportion. Form of the Sampling Distribution of p p The sampling distribution of can be approximated by a normal distribution whenever the sample size is large. The sample size is considered large whenever these conditions are satisfied:
np > 5 and n(1 – p) > 5 Form of the Sampling Distribution of p For values of p near .50, sample sizes as small as 10
permit a normal approximation.
With very small (approaching 0) or very large (approaching 1) values of p, much larger samples are needed. Sampling Distribution of p
s Example: St. Andrew’s College Recall that 72% of the
prospective students applying
to St. Andrew’s College desire
oncampus housing. What is the probability that
a simple random sample of 30 applicants will provide
an estimate of the population proportion of applicant
desiring oncampus housing that is within plus or
minus .05 of the actual population proportion? Sampling Distribution of p
For our example, with n = 30 and p = .72, the normal distribution is an acceptable approximation because:
np = 30(.72) = 21.6 > 5
and
n(1 p) = 30(.28) = 8.4 > 5 Sampling Distribution of p Sampling
Distribution
p
of σp = E( p ) = .72 .72(1 − .72)
= .082
30 p Sampling Distribution of p
Step 1: Calculate the zvalue at the upper endpoint of the interval.
z = (.77 − .72)/.082 = .61
Step 2: Find the area under the curve to the left of the upper endpoint.
P(z < .61) = .7291 Sampling Distribution of p
Cumulative Probabilities for the Standard Normal Distribution
z .00 .01 .02 .03 .04 .05 .06 .07 .08 .09 . . . . . . . . . . . .5 .6915 .6950 .6985 .7019 .7054 .7088 .7123 .7157 .7190 .7224 .6 .7257 .7291 .7324 .7357 .7389 .7422 .7454 .7486 .7517 .7549 .7 .7580 .7611 .7642 .7673 .7704 .7734 .7764 .7794 .7823 .7852 .8 .7881 .7910 .7939 .7967 .7995 .8023 .8051 .8078 .8106 .8133 .9 .8159 .8186 .8212 .8238 .8264 .8289 .8315 .8340 .8365 .8389 . . . . . . . . . . . Sampling Distribution of p Sampling
Distribution
p
of σ p = .082 Area = .7291 p
.72 .77 Sampling Distribution of p
Step 3: Calculate the zvalue at the lower endpoint of the interval.
z = (.67 − .72)/.082 = .61
Step 4: Find the area under the curve to the left of the lower endpoint.
P(z < .61) = .2709 Sampling Distribution of p Sampling
Distribution
p
of σ p = .082 Area = .2709 p
.67 .72 Sampling Distribution of p
Step 5: Calculate the area under the curve between the lower and upper endpoints of the interval.
P(.61 < z < .61) = P(z < .61) − P(z < .61)
= .7291 − .2709
= .4582
The probability that the sample proportion of applicants
wanting oncampus housing will be within +/.05 of the
actual population proportion : p
P(.67 < < .77) = .4582 Sampling Distribution of p Sampling
Distribution
p
of σ p = .082 Area = .4582 .67 .72 .77 p Other Sampling M ethods
•
•
•
•
• Stratified Random Sampling
Cluster Sampling
Systematic Sampling
Convenience Sampling
Judgment Sampling Stratified Random Sampling The population is first divided into groups of elements called strata. Each element in the population belongs to one and only one stratum. Best results are obtained when the elements within each stratum are as much alike as possible (i.e. a homogeneous group). Stratified Random Sampling A simple random sample is taken from each stratum. Formulas are available for combining the stratum sample results into one population parameter estimate. Advantage: If strata are homogeneous, this method is as “precise” as simple random sampling but with a smaller total sample size. Example: The basis for forming the strata might be department, location, age, industry type, and so on. Cluster Sampling The population is first divided into separate groups of elements called clusters. Ideally, each cluster is a representative smallscale version of the population (i.e. heterogeneous group). A simple random sample of the clusters is then taken. All elements within each sampled (chosen) cluster form the sample. Cluster Sampling Example: A primary application is area sampling, where clusters are city blocks or other welldefined areas. Advantage: The close proximity of elements can be cost effective (i.e. many sample observations can be obtained in a short time). Disadvantage: This method generally requires a larger total sample size than simple or stratified random sampling. Systematic Sampling If a sample size of n is desired from a population containing N elements, we might sample one element for every n/N elements in the population. We randomly select one of the first n/N elements from the population list. We then select every n/Nth element that follows in the population list. Systematic Sampling This method has the properties of a simple random sample, especially if the list of the population elements is a random ordering. Advantage: The sample usually will be easier to identify than it would be if simple random sampling were used. Example: Selecting every 100th listing in a telephone book after the first randomly selected listing Convenience Sampling It is a nonprobability sampling technique. Items are included in the sample without known probabilities of being selected. The sample is identified primarily by convenience. Example: A professor conducting research might use student volunteers to constitute a sample. Convenience Sampling Advantage: Sample selection and data collection are relatively easy. Disadvantage: It is impossible to determine how representative of the population the sample is. Judgment Sampling The person most knowledgeable on the subject of the study selects elements of the population that he or she feels are most representative of the population. It is a nonprobability sampling technique. Example: A reporter might sample three or four senators, judging them as reflecting the general opinion of the senate. Judgment Sampling Advantage: It is a relatively easy way of selecting a sample. Disadvantage: The quality of the sample results depends on the judgment of the person selecting the sample. ...
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