Transcript Sampling and Sampling Distributions •Simple Random Sampling •Point Estimation

Sampling and Sampling
Distributions
•Simple Random Sampling
•Point Estimation
•Sampling Distribution
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:
Finite Population
• Finite populations are often defined by lists such
as:
– Organization membership roster
– Credit card account numbers
– Inventory product numbers
A Good Sample is a Random Sample
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:
Finite 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, computer-generated
random numbers are often used to automate the
sample selection process.
Simple Random Sampling:
Infinite 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.

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:
Infinite 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.
We refer to
mean .
x as the point estimator of the population
s is the point estimator of the population standard
deviation .
p is the point estimator of the population proportion p.
Sampling Error
 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

The sampling errors are:
|x   | for sample mean
|s   | for sample standard deviation
| p  p| for sample proportion
Example: St. Andrew’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 on-campus housing.
Example: St. Andrew’s
We will now look at three
alternatives for obtaining the
desired information.
• Conducting a census of the
entire 900 applicants
• Selecting a sample of 30
applicants, using a random number table
• Selecting a sample of 30 applicants, using Excel
Conducting a Census

If the relevant information for the entire 900 applicants
was 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


i
900
 990
• Population Standard Deviation for SAT Score

2
(
x


)
 i
900
 80
• Population Proportion Wanting On-Campus
Housing
648
p
 .72
900
Simple Random Sampling
 Now suppose that the necessary information on
the current year’s applicants was 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:
Using a Random Number Table
• Taking a Sample of 30 Applicants
• Since the finite population has 900 elements, we
will need 3-digit random numbers to randomly
select applicants numbered from 1 to 900.
• We will use the last three digits of the 5-digit
random numbers in the third column of the
textbook’s random number table, and continue
into the fourth column as needed.
(See Table 7.1, p. 275)
Simple Random Sampling:
Using a Random Number Table
• Taking a Sample of 30 Applicants
•
The numbers we draw will be the numbers of the
applicants we will sample unless
• the random number is greater than 900 or
• the random number has already been used.
• We will continue to draw random numbers until
we have selected 30 applicants for our sample.
•
(We will go through all of column 3 and part of
column 4 of the random number table, encountering
in the process five numbers greater than 900 and
one duplicate, 835.)
Use of Random Numbers for Sampling
See 3rd column of Table 7.1, p. 275
3-Digit
Applicant
Random Number Included in Sample
744
No. 744
436
No. 436
865
No. 865
790
No. 790
835
No. 835
902
Number exceeds 900
190
No. 190
836
No. 836
. . . and so on
Simple Random Sampling:
Using a Random Number Table
• Sample Data
Random
No. Number
1
744
2
436
3
865
4
790
5
835
.
.
.
.
30
498
Applicant
Conrad Harris
Enrique Romero
Fabian Avante
Lucila Cruz
Chan Chiang
.
.
SAT
Score
1025
950
1090
1120
930
.
.
Live OnCampus
Yes
Yes
No
Yes
No
.
.
Emily Morse
1010
No
Using Excel to Select
a Simple Random Sample
• Taking a Sample of 30 Applicants
• Excel provides a function for generating random
numbers in its worksheet.
• 900 random numbers are generated, one for each
applicant in the population.
• Then we choose the 30 applicants corresponding
to the 30 smallest random numbers as our sample.
• Each of the 900 applicants has the same probability
of being included.
Using Excel to Select
a Simple Random Sample
• Formula Worksheet
1
2
3
4
5
6
7
8
9
A
B
C
Applicant
Number
1
2
3
4
5
6
7
8
SAT
Score
1008
1025
952
1090
1127
1015
965
1161
On-Campus
Housing
Yes
No
Yes
Yes
Yes
No
Yes
No
Note: Rows 10-901 are not shown.
D
Random
Number
=RAND()
=RAND()
=RAND()
=RAND()
=RAND()
=RAND()
=RAND()
=RAND()
Using Excel to Select
a Simple Random Sample

Value Worksheet
A
1
2
3
4
5
6
7
8
9
B
C
Applicant
SAT
On-Campus
Number
Score
Housing
1
1008
Yes
2
1025
No
3
952
Yes
4
1090
Yes
5
1127
Yes
6
1015
No
7
965
Yes
8
1161
No
Note: Rows 10-901 are not shown.
D
Random
Number
0.84891
0.30181
0.35709
0.99433
0.33072
0.54548
0.46758
0.33167
Using Excel to Select
a Simple Random Sample

Put Random Numbers in Ascending Order
Step 1 Select cells A2:A901
Step 2 Select the Data menu
Step 3 Choose the Sort option
Step 4 When the Sort dialog box appears:
Choose Random Numbers in the
Sort by text box
Choose Ascending
Click OK
Using Excel to Select
a Simple Random Sample
• Value Worksheet (Sorted)
1
2
3
4
5
6
7
8
9
A
B
C
D
Applicant
Number
12
773
408
58
116
185
510
394
SAT
Score
1107
1043
991
1008
1127
982
1163
1008
On-Campus
Housing
No
Yes
Yes
No
Yes
Yes
Yes
No
Random
Number
0.00027
0.00192
0.00303
0.00481
0.00538
0.00583
0.00649
0.00667
Note: Rows 10-901 are not shown.
Point Estimates
•
as Point Estimator of 
x

x
29, 910

 997
30
30
•
s as Point Estimator of 
s
p•
i
2
(
x

x
)
 i
29

x
163, 996
 75.2
29
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
Parameter
Value
 = Population mean
990
x = Sample mean
997
 = Population std.
80
s = Sample std.
deviation for
SAT score
75.2
p = Population proportion wanting
campus housing
.72
p = Sample pro-
.68
SAT score
deviation for
SAT score
Point
Estimator
Point
Estimate
SAT score
portion wanting
campus housing
Sampling Distribution of x
• Process of Statistical Inference
Population
with mean
=?
The value of x is used to
make inferences about
the value of .
A simple random sample
of n elements is selected
from the population.
The sample data
provide a value for
the sample mean x.
Sampling Distribution of x
The sampling distribution of x
is the probability distribution of
all possible values of the
sample mean
x
x
E (x )  
where:
 = the population mean
Sampling Distribution of x
Standard Deviation of x
Finite Population

Infinite Population
N n
x  ( )
n N 1
x 

n
• A finite population is treated as being
infinite if n/N < .05.
•
( N  n) / ( N  1) is
the finite correction factor.
•  x is referred to as the standard error of the
mean.
Sampling Distribution of x
If we use a large (n > 30) simple random sample, the
central limit theorem enables us to conclude that the
sampling distribution of x can be approximated by
a normal probability distribution.
When the simple random sample is small (n < 30),
the sampling distribution of x can be considered
normal only if we assume the population has a
normal probability distribution.
Central Limit Theorem
In selecting simple random samples
of size n, the sampling distribution of
the mean can be approximated by a
normal probability distribution as the
sample size becomes large.
Sampling Distribution of x for SAT Scores
Sampling
Distribution
of x
E( x )  990
x 

80

 14.6
n
30
x
Sampling Distribution of x 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  ?
In other words, what is the probability that x will be
between 980 and 1000?
Sampling Distribution of x for SAT Scores
Step 1: Calculate the z-value 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
Sampling Distribution of x 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
.
.
.
.
.
.
.
.
.
.
.
Sampling Distribution of x for SAT Scores
Sampling
Distribution
of x
 x  14.6
Area = .7517
x
990 1000
Sampling Distribution of x for SAT Scores
Step 3: Calculate the z-value 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) = P(z > .68)
= 1 - P(z < .68)
= 1 - . 7517
= .2483
Sampling Distribution of x for SAT Scores
Sampling
Distribution
of x
 x  14.6
Area = .2483
x
980 990
Sampling Distribution of x 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:
P(980 <
x < 1000) = .5034
Sampling Distribution of x for SAT Scores
Sampling
Distribution
of x
 x  14.6
Area = .5034
980 990 1000
x