Chapter 7, Part A Sampling and Sampling Distributions

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Chapter 7, Part ASampling and Sampling
Distributions

• Simple Random Sampling

• Point Estimation

• Introduction to Sampling Distributions

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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.
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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.
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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.

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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.

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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.

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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.

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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 ?.
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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.

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Sampling Error

• The sampling errors are

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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.

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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.

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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

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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.

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Conducting a Census

• Population Mean SAT Score

• Population Standard Deviation for SAT Score

• Population Proportion Wanting On-Campus Housing

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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.

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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.

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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.)

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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
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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
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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.

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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.
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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
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• Process of Statistical Inference

A simple random sample of n elements is
selected from the population.
Population with mean m ?
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where ? the population mean
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Finite Population
Infinite Population

• A finite population is treated as being
• infinite if n/N lt .05.

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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
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Cumulative Probabilities for the Standard Normal
Distribution
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Area .7517
990
1000
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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
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Area .2483
980
990
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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
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Area .5034
1000
980
990
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• Suppose we select a simple random sample of
  100
• applicants instead of the 30 originally
  considered.

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Area .7888
1000
980
990