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Chapter 7, Part A Sampling and Sampling Distributions

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Title: Chapter 7, Part A Sampling and Sampling Distributions


1
Chapter 7, Part ASampling and Sampling
Distributions
  • Simple Random Sampling
  • Point Estimation
  • Introduction to Sampling Distributions

2
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.
3
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.
4
Simple Random SamplingFinite Population
  • Finite populations are often defined by lists
    such as
  • Organization membership roster
  • Credit card account numbers
  • Inventory product numbers
  • 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.

5
Simple Random SamplingFinite 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.

6
Simple Random SamplingInfinite 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.

7
Simple Random SamplingInfinite 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.

8
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.
s is the point estimator of the population
standard deviation ?.
9
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.

10
Sampling Error
  • The sampling errors are

11
Example St. Andrews
  • St. Andrews College receives
  • 900 applications annually from
  • prospective students. The
  • application form contains
  • a variety of information
  • including the individuals
  • scholastic aptitude test (SAT) score and whether
    or not
  • the individual desires on-campus housing.

12
Example St. Andrews
  • 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.

13
Example St. Andrews
  • 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

14
Conducting a Census
  • If the relevant data for the entire 900
    applicants were in the colleges 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.

15
Conducting a Census
  • Population Mean SAT Score
  • Population Standard Deviation for SAT Score
  • Population Proportion Wanting On-Campus Housing

16
Simple Random Sampling
  • Now suppose that the necessary data on the
  • current years applicants were not yet
    entered in the
  • colleges 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.

17
Simple Random SamplingUsing a Random Number
Table
  • Taking a Sample of 30 Applicants
  • Because 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
  • textbooks random number table, and continue
  • into the fourth column as needed.

18
Simple Random SamplingUsing 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.)

19
Simple Random SamplingUsing a Random Number
Table
  • Use of Random Numbers for Sampling

3-Digit Random Number
Applicant 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
20
Simple Random SamplingUsing a Random Number
Table
  • Sample Data

Random Number
SAT Score
Live On- Campus
No.
Applicant
1 744 Conrad Harris 1025 Yes
2 436 Enrique Romero 950 Yes
3 865 Fabian Avante 1090 No
4 790 Lucila Cruz 1120 Yes
5 835 Chan Chiang 930 No
. . . . .
. . . . .
30 498 Emily Morse 1010 No
21
Simple Random SamplingUsing a Computer
  • Taking a Sample of 30 Applicants
  • Computers can be used to generate random
  • numbers for selecting random samples.
  • For example, Excels function
  • RANDBETWEEN(1,900)
  • can be used to generate random numbers
    between
  • 1 and 900.
  • Then we choose the 30 applicants corresponding
  • to the 30 smallest random numbers as our
    sample.

22
Point Estimation
  • s as Point Estimator of ?

Note Different random numbers would
have identified a different sample which would
have resulted in different point estimates.
23
Summary of Point Estimates Obtained from a Simple
Random Sample
Population Parameter
Point Estimator
Point Estimate
Parameter Value
m Population mean SAT score
990
997
80
s Sample std. deviation for SAT
score
75.2
s Population std. deviation for
SAT score
.72
.68
p Population pro- portion wanting
campus housing
24
  • Process of Statistical Inference


A simple random sample of n elements is
selected from the population.
Population with mean m ?
25
where ? the population mean
26
Finite Population
Infinite Population
  • A finite population is treated as being
  • infinite if n/N lt .05.

27
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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 lt .68) .7517
31
Cumulative Probabilities for the Standard Normal
Distribution
32
Area .7517
990
1000
33
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 lt -.68) P(z gt .68)
1 - P(z lt .68)
1 - . 7517
.2483
34
Area .2483
980
990
35
Step 5 Calculate the area under the curve
between the lower and upper endpoints of the
interval.
P(-.68 lt z lt .68) P(z lt .68) - P(z lt -.68)
.7517 - .2483
.5034
The probability that the sample mean SAT score
will be between 980 and 1000 is
36
Area .5034
1000
980
990
37
  • Suppose we select a simple random sample of
    100
  • applicants instead of the 30 originally
    considered.

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40
Area .7888
1000
980
990
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