Sampling Methods and Sampling Distributions

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Sampling Methods and Sampling Distributions

• Potential sampling errors
• Sampling Distributions and the Central Limit
  Theorem
• Confidence Intervals
• Review

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Review of terms

• A target population is the entire group of
  elements about which we want information.
• A sample is part of the target population.

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Inference

• Making an inference means using sample results to
  describe the population.

Sample (Known)
Population (Unknown)
We dont know the mean of the population so we
have to infer it from samples of the population
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Sampling Questions

• What errors might there be in a sample conducted
  over the phone?
• If you wanted to estimate the number of people
  who would vote Liberal if an election was held
  tomorrow, how would you go about it?

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Sampling Terminology

• An Element is an object on which we take a
  Measurement. Objects that are people are called
  Subjects.
• A Target Population is a collection of elements
  about which we wish to make an Inference.
• Sampling Units are non-overlapping collections
  of elements from the target population.
• A Frame is a list of sampling units.
• The sampling Design specifies the Method of
  selecting the sample.

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Errors in Survey Sampling

• Selection Error
• Sampling frame does not represent target
  population. We exclude members of the target
  population from the sample.
• Interested in determining filmgoers attitudes
  toward horror films. Sampling frame is households
  that own a VCR. Many filmgoers do not own VCRs.
  We have committed the selection error.
• Increasing the Sample Size Will Not Help.

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Errors in Survey Sampling

• Response Error
• Respondents do not
• 1) Understand question
• 2) Have the information
• 3) Want to give the information.
• Ask 13 year old school students the following
  question How often do you imbibe intoxicating
  spirits? Respondents may not understand or be
  honest.
• Increasing the Sample Size Will Not Help.

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Errors in Survey Sampling

• Non-Response Error
• Respondents are not representative of sampling
  frame.
• Be concerned when a large percentage of the
  sampling frame does not respond.
• Lower income families may ignore mailed surveys.
• Families with two wage earners eat out often and
  are often not at home when an interviewer calls.
• Increasing the Sample Size Will Not Help

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More terms Parameters and Statistics

• A population parameter is a numerical measure
  that describes the target population.
• A sample statistic is an estimate of the unknown
  population parameter and will vary from sample to
  sample.

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A small population (N 5)
Number of bedrooms per household 1 2 2 3 5
Note that the denominator for the standard
deviation calculation is N 5 because this is a
population
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A single sample of size n 2 from the population
of N 5

• Do not expect
• Sample mean to equal population mean of 2.6.
• Sample standard deviation to equal
• population standard deviation of 1.36.

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Sample Statistics

• Note that the sample standard deviation has a
  different formula than the population standard
  deviation
• To help keep the ideas separate we have different
  symbols for populations and samples
• The sample mean is
• The sample standard deviation is s

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Margin of Error

• Because samples statistics and population
  parameters are inevitably (usually) going to be
  different we have some error when we take a
  sample.
• But what affects the amount of error?
• Dartboard example

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Margins of Error
Margin of Error Possible difference between the
sample result and the result we would obtain if
we selected the entire population. Want as small
as possible.
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Samples of 3 from population
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Effect of sample size
Samples of Size 3
Samples of Size 4
2.6
2.6
Population Mean
3.33 1.67
3 2
Largest and Smallest Sample Means
Maximum Margin of Error
0.93
0.6
Increasing the sample size reduces the margin of
error
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Effect of level of confidence
Increasing the level of confidence increases the
margin of error
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Effect of population variance
New Population 1 2 4 6 7 bdrms
As the variance of the population increases the
margin of error also increases
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SummarySampling Lessons

• Increasing the sample size reduces the margin
  of error.
• If we increase the level of confidence in an
  inference, the price we pay is in the margin of
  error.
• As the variability of the target population
  increases, the margin of error increases.

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Sampling Distribution

• What is a sampling distribution of the mean?

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Bedrooms (samples of n 3)
The sampling distribution contains all
possible sample means.
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Sampling distribution
This is a sampling distribution
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Standard Error of the Mean

• The standard deviation of the sampling
  distribution measures the spread of the sample
  means around their mean and is called the
  standard error of the mean.
• The standard error of the mean is smaller than
  the standard deviation of the population.
• Why?

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2 New Populations (both N6)

• A 1, 1, 2, 4, 5, 5
• B 1, 2, 3, 3, 4, 5

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Central Limit Theorem

• No matter what the population distribution looks
  like, the sampling distribution of the mean will
  always end up looking like a normal distribution
  (for high enough n).

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Try playing with the Central Limit Theorem on the
class web page. - Try different sample sizes
(n). - Try different population distributions.
- See how the sampling distributions look
normal.
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Some Conclusions
Population
Sampling Distribution
Mean
(unknown)
Standard Deviation
(unknown)
Shape
Any Shape
Approx Normal provided n gt 30
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Estimating Unknown Population Parameters
Unknown Parameter
Sample Statistic
Mean
Standard Deviation
s
Standard Error
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Why does the Central Limit Theorem work?

• As sample size increases
• most sample means will be close to population
  mean.
• some sample means will be relatively far above or
  below population mean.
• a few sample means will be very far above or
  below population mean.
• Above bullets describe a normal distribution.

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Lessons

• The mean of any distribution of the sample mean
  is the same as the mean of the population from
  which it was derived.
• The standard error of the mean is smaller than
  the standard deviation of the population.

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Lessons

• The standard error of the mean decreases as the
  sample size increases.
• If the population is normal or the sample size is
  sufficiently large, the distribution of the
  sample mean will be near-normal. We will be able
  to use the standard normal table to compute
  probabilities for the sample means.

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Two assumptions for Central Limit Theorem to work

• 1) Samples are drawn randomly from population
  (each possible sample has an equal chance of
  being chosen)
• 2) The population is (near) normal or the sample
  size is large (n ? 30)

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Overview of Inference
Draw Conclusion about a Population Parameter
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Confidence Interval

• A confidence interval is a range estimate of an
  unknown population parameter.
• The level of confidence associated with an
  interval estimate is the percentage of intervals
  that will include the unknown population over a
  large number of similarly constructed intervals.
• Just like the confidence we had in margin of
  error in an earlier lecture (dartboard example)

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Confidence Intervals
Sampling Distribution of the mean
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What does 95 confidence look like? (a 0.05)
Each probability 0.025
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Intervals and Confidence Level
Confidence Intervals
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Margin of Error

• So what a confidence interval does is add and
  subtract a margin of error from the sample mean
• The margin of error is
• but if we dont know ? then well have to use s
  (the sample standard deviation) instead.

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Assumptions for confidence intervals

• 1. Random samples
• 2. If n lt 30 then population must be near normal
  to do a confidence interval. (If n ? 30 then
  sampling distribution is close enough to normal
  whatever the population.)

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Margin of Error - Three Lessons
Lesson 1 As sample size (n) increases,
margin of error decreases. Lesson 2 As
confidence level increases (z),
margin of error increases. Lesson 3 As variance
increases (s2), margin of error
increases.
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Rules of thumb

• Some quick rules of thumb for z (values come from
  normal distribution)
• For confidence of 90 use z 1.64
• For confidence of 95 use z 2
• For confidence of 99 use z 2.58

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T-distribution

• What is a t-value?
• Sophisticated statistical way of dealing with
  smaller sample sizes by using slightly different
  values instead of the rule of thumb z-values
• Do I have to care?
• No. Increased accuracy of t-values possibly
  spurious and not worth the added effort. Just
  think z-value wherever you see t-value and use
  rule of thumb
• If accuracy is important, can I use the t-value
  anyway?
• Yes. Statpro calculations automatically use
  t-values.

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Width versus meaningfulness of Confidence
Intervals
GOAL Narrow Confidence Interval and high
level of confidence.
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Try the confidence interval demonstration on the
class web page. Try different values of ?. Count
how many of the confidence intervals contain the
population mean.
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Using Statpro

• Make sure data is in a column with label in first
  row.
• Use Statpro function
• Statistical Inference gt One sample analysis
• Select data
• Choose confidence interval for mean and input
  confidence level (e.g. 95)

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Question

• As the sample size increases, does the estimated
  standard error increase, decrease, or stay the
  same?

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Question

• As the sample size increases, does the sample
  standard deviation increase, decrease, or stay
  the same?

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Question

• As the sample size increases, does the sample
  mean increase, decrease, or stay the same?

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Question

• As the sample size increases, does the margin of
  error increase, decrease, or stay the same?

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Second hand cars

• In a survey of their latest 20 customers, a
  second hand car dealer found that the average age
  of car buyers is 37.3 years old with a standard
  deviation of 4.2 years.
• What is a 95 CI for the mean age of secondhand
  car buyers?

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Small populations

• If you have a relatively large sample compared to
  the population (n/N gt 0.05)
• Use correction for confidence interval

N number in population n number in sample
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What did we do?

• Talked about margins of error.
• Saw how the Central Limit Theorem ensures that
  means always have a normal distribution.
• Talked about confidence intervals
• Reviewed first half of material for subject

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Managerial applications

• What did you learn today that makes a difference
  to the way you manage?
• What are the three most important things to
  remember from todays lecture?

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Next lecture (after Midterm)

• Download data file metrobus.xls and
  customerages.xls and bring them on laptop
• Read supplementary material on Two Samples,
  Matched Pairs and Estimating P.