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Chapter 7 Sampling and Sampling Distribution
In review…
-- The purpose of statistical inference is to make estimates and test hypothesis about the characteristics of the population from
info contained in a sample.
-- A population is the set of all the elements of interest
-- A sample is a subset of the population
-- 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.
7.2 Simple Random Sampling
Sampling from a finite 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.
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.
 Excel has random number generating functions RAND & RANDBETWEEN
Sampling from a 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  avoid selection bias
-- 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.
7.3 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 is the point estimator of the population mean .
s is a point estimator of the population standard deviation .
p is a 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.
Sampling errors are:
For sample mean  |x   |
For sample standard deviation  |s   |
For sample proportions  |
p-p|
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Example for Sections 7.2/7.3 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.
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
We will now look at three alternatives for obtaining the desired information.
1) Conducting a census of the entire 900 applicants
2) Selecting a sample of 30 applicants, using a random number table
3) Selecting a sample of 30 applicants, using Excel
Conducting a Census
-- 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.
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 then decides a sample of 30 applicants will be used.
-- The applicants were numbered, from 1 to 900, as their applications arrived.
A) Using a Random Number Table
-- 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
textbook’s random number table (p 269), and continue into the fourth column as needed.
-- 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.)
3 Digit Random #
744
436
865
790
835
902
190
836
… and so on
Applicant Included in Sample
#744
#436
#865
#790
#835
exceeds 900
#190
#836
2
No.
1
2
3
4
5
.
.
.
30
Random Number
744
436
865
790
835
.
.
.
498
Applicant
SAT Score
Conrad Harris
1025
Enrique Romero
950
Fabian Avante
1090
Lucila Cruz
1120
Chan Chiang
930
.
.
.
.
.
.
Emily Morse
1010
Live On Campus
Yes
Yes
No
Yes
No
.
.
.
No
B) Using Excel
-- Excel’s function = RANDBETWEEN(1,900) can be used to generate random numbers between
1 and 900.
-- Choose the 30 applicants corresponding to the 30 smallest random numbers as our sample.
Point estimator of μ
Point estimator of σ
Point estimator of p
Summary of Point Estimates
Population Parameter
Parameter Value
Point Estimator
μ = Population mean
SAT score
990
x = sample mean
SAT score
σ = Population Std
Deviation for
SAT score
80
s
p = population proportion
wanting campus housing
.72
p
3
Point Estimate
997
75.2
.68
7.4/7.5 Sampling Distribution of x
--The sampling distribution of x is the probability distribution of all possible values of the sample mean x
Expected Value of x
E(x )=μ
where μ is the population mean
Standard Deviation of x
Finite Population
 xx  (

n
)
Infinite Population
N n
N 1
 xx 

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 the standard error of the mean
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 distribution.
Central Limit Theorem
In selecting simple random samples of size n from a population, the sampling distribution of the sample mean x can be
approximated by a normal distribution as the sample size becomes large
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 distribution.
Example: 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?
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Step 1: Calculate the z values (in this example, at the upper and lower endpoints of the interval)
Step 2: Shade the area under the curve corresponding to the region you are solving for (can include both a normal and standard
normal distribution)
Step 3: Calculate the area under the curve you are solving for (state the cumulative probabilities or what you know first)
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Relationship Between the Sample Size and the Sampling Distribution of x
-- Suppose we select a simple random sample of 45 applicants instead of the 30 originally considered
-- E( x ) =μ regardless of the sample size. In our example, E( x ) remains at 990.
-- Whenever the sample size is increased, the standard error of the mean x is decreased. With the increase
in the sample size to n = 45, the standard error of the mean (  x ) has decreased to:
-- Recall that when n = 30, P(980 < x < 1000) = .5034.
-- We follow the same steps to solve for P(980 < x < 1000) when n = 45 as we showed earlier when n = 30.
-- Now, with n = 45, P(980 < x < 1000) = __________
Step 1: Calculate the z values (in this example, at the upper and lower endpoints of the interval)
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Step 2: Shade the area under the curve corresponding to the region you are solving for (can include both a normal and standard
normal distribution)
Step 3: Calculate the area under the curve you are solving for (state the cumulative probabilities or what you know first)
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Why? Mathematically, because
 x is in the denominator of the Z score calculation and as  x increase, Z score
decreases and vice versa
z 
x 
 xx
-- Because the sampling distribution with n = 100 has a smaller standard error, the values of x
variability and tend to be closer to the population mean than the values of x with n = 30.
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