Download Population and Sampling Distribution

Chapter Twelve
12-1
Sample Size for Probability Sampling
Census:
• Population canvass - not really a “sample”
• Asking the entire population
Budget Available:
• A valid factor – how much can we afford?
Rule of Thumb:
• Is there some convention we can apply?
• What might make an adequate sample size?
Key Terms & Definitions
12-2
Sample Size for Probability Sampling
Number of Subgroups Analyzed:
In any sample size determination problem, consideration
must be given to the number and anticipated size of
various subgroups of the total sample that must be
analyzed.
Traditional Statistical Methods:
Variance, standard deviation, and confidence interval
play a key role.
Key Terms & Definitions
12-3
Sample Size for Nonprobability Sampling
Judgment:
• Best guess of “experts”
• Draw on your experience to determine sample
size
Conventional:
• What have others done?
• See what the sample size has been for similar
studies
Key Terms & Definitions
12-4
The Normal Distribution
Central Limit Theorem:
The idea that a distribution of a large number of sample means
or sample proportions will approximate a normal distribution,
regardless of the distribution of the population from which they
were drawn.
Normal Distribution:
The continuous distribution that is bell shaped and symmetrical
about the mean. The mean, median, and mode are equal.
Finally, about 68% of the observations are within one standard
deviation plus/minus of the mean, 96% are within two standard
deviations, and 99+% are within three standard deviations of
the mean respectively.
Key Terms & Definitions
12-5
The Normal Distribution
Proportionate Properties of the Normal Distribution:
A feature that the number of observations falling
between the mean and a given number of standard
deviations from the mean is the same for all normal
distributions.
Key Terms & Definitions
12-6
The Normal Distribution
Standard Normal Distribution:
Normal distribution with a mean of zero and a
standard deviation of one.
Key Terms & Definitions
12-7
The Normal Distribution
Standard Deviation:
Standard
Deviation
Key Terms & Definitions
=
sum
The measure of dispersion calculated by subtracting the mean of
the series from each value in a series, squaring each result,
summing the results, dividing the sum by the number of
observations minus 1 and finally taking the square root of this
value.
(X1- X)
2
(N-1)
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Important Characteristics of
a Normal Distribution
1. Bell-shaped and has only one mode.
2. Symmetric about its mean.
3. Uniquely defined by its mean and standard deviation.
4. The total area under a normal curve is equal to one.
5. The area of a region under the normal distribution curve
between any two values of a variable equals the probability of
observing a value in that range when an observation is
randomly selected from the distribution.
6. The area between the mean and a given number of standard
deviations from the mean is the same for all normal
distributions.
Key Terms & Definitions
12-8
Population and Sampling Distribution
Population Distribution:
The frequency distribution of all the elements of a population.
Sampling Distribution:
The frequency distribution of all the elements of an individual
sample.
Key Terms & Definitions
12-10
Sampling Distribution of the Mean
Sampling Distribution of the Mean:
The theoretical frequency distribution of the means of all possible
samples of a given size drawn from a particular population; it is
normally distributed.
Key Terms & Definitions
12-11
Standard Error of the Mean
Standard Error of the Mean:
Standard deviation of a distribution of sample means.
Key Terms & Definitions
12-12
Relationships of the
Three Basic Types of Distribution
Key Terms & Definitions
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Simple Random Sample
The results of a simple random sample of
fast-food restaurant patrons could be used to
compute the mean number of visits for the
period of 1 month for 1,000 samples.
Key Terms & Definitions
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Simple Random Sample
Key Terms & Definitions
12-15
Point and Interval Estimations
Point Estimate:
The particular estimate of a population value.
Interval Estimate:
The interval or range of values within which the true population
value is estimated to fall.
Key Terms & Definitions
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Confidence Level and Interval
Confidence Level:
The probability that a particular interval will include the true
population value; also called the confidence coefficient.
Confidence Interval:
The interval that, at the specified confidence level, includes
the true population value.
Key Terms & Definitions
12-17
Sampling Distribution of the Proportion
Sampling Distribution of the Proportion:
The relative frequency distribution of the sample
proportions of many random samples of a given size drawn
from a particular population; it is normally distributed.
Key Terms & Definitions
12-18
Sampling Distribution of the Proportion
Examples:
• The percentage of a population is aware of a
particular ad.
• The percentage of the population that accesses the
Internet one or more times in an average week.
• The percentage of the population that has visited a
fast-food restaurant four or more times in the past
30 days.
• The percentage of the population that watches a
particular television program.
Key Terms & Definitions
12-19
Determining the Sampling Size
Problems Involving Mean:
Key Terms & Definitions
12-17
Determining the Sampling Size
1. Allowable Sampling Error:
Amount of sampling error the researcher is willing to accept, E.
2. The Acceptable Level of Confidence:
How confident does the researcher want to be that an
interval includes the population mean, Z.
3. Population Standard Deviation:
Standard deviation of a variable for the entire population, σ.
Key Terms & Definitions
12-17
Determining the Sampling Size
Determining Population Standard Deviation:
How to determine before selecting the sample:
•
•
•
•
Use results from a prior survey
o In many cases, your firm may have conducted a prior survey
dealing with similar subject matter.
Conduct a pilot survey
o For a large-scale project, you may conduct a smaller-scale pilot to
help determine your standard deviation.
Use secondary data
o Using secondary resources to accumulate data about your
population sample.
Use judgment
o When all else fails, estimating it based on your educated judgment.
Key Terms & Definitions
12-22
Determining the Sampling Size
Problems Involving Proportions:
Key Terms & Definitions
12-23
Population Size and Sampling Size
Independence Assumption:
Assumption that sample elements are drawn independently.
Key Terms & Definitions
12-24
Errors
Statistical Errors:
Most of the formulas in this unit focus on
determining type I errors or the error that
concludes there is a difference when there is
no actual difference.
A type II error is one where you say there is
no difference when there is an actual
difference. The probability of not making this
type of error is called statistical power.
Key Terms & Definitions
12-25
Finite Population Correction Factor (FPC)
Statistical Power:
The probability of not making a type II error.
Key Terms & Definitions
12-26
How Many Contacts?
Determining the number of phone numbers you’ll need to connect
with the sample size you desire requires:
•
The number of completed interviews you need.
•
The working phone or “reachable” rate.
•
The incidence rate or the number of the people you call who
will answer and qualify for the interview.
•
The contact/completion rate which includes:
o Contact rate
o Length of interview
o Sensitivity of topic
o Time of year
o Number of callbacks
Key Terms & Definitions
12-27
How Many Contacts?
You need to contact more than n, since it’s unlikely that everyone
you contact will agree to answer the questionnaire. If you
determine that you need 800 respondents and a typical response
rate for your type of survey is 30%, then you would actually have to
contact 2,667 people to get 800 completed surveys.
Number of Contacts = n/.30
OR
800/.3
Number of Contacts = 2,667
You must contact approximately 2,667 people (with
30% response rate), so you can expect 800 people to
respond (the final sample size).
Key Terms & Definitions
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Sampling Issues To Consider
• Time to Generate the Sample
• Scope of the Research
• Budget Available
• Experience with Sampling
• Level of Accuracy Desired
• Your Knowledge of the Population
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Key Terms & Definitions
• Central Limit Theorem
• Normal Distribution
• Proportional Property of the
Normal Distribution
• Standard Normal Distribution
• Sample Size for
Nonprobability Sampling
• Standard Deviation
• Population Distribution
• Sample Distribution
• Sampling Distribution of the Mean
Links and
Key Terms & Definitions
• Standard Error of the Mean
• Point Estimate
• Interval Estimate
• Confidence Level
• Confidence Interval
• Sampling Distribution of the Proportion
• Allowable Sampling Error
• Population Standard Deviation
• Finite Population Correction Factor (FPC)
• Statistical Power
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