Estimation in Sampling

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Estimation in Sampling

• GTECH 201
• Lecture 15

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Conceptual Setting

• How do we come to conclusions from empirical
  evidence?
• Isnt common sense enough?
• Why?
• Systematic methods for drawing conclusions from
  data
• Statistical inference
• Inductive versus Deductive Reasoning

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Drawing Conclusions

• Statistical inference
• Based on the laws of probability
• What would happen if?
• You ran your experiment hundreds of times
• You repeated your survey over and over again
• Statistic and Parameter
• The proportion of the population who are
  ltdisabledgt usually denoted by p
• In a SRS of 1000 people, the proportion of the
  people who are ltdisabledgt usually denoted by
  (p -hat)

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Estimating with Confidence

• Say you are conducting an opinion poll
• SRS of 1000 adult television viewers
• You ask these folks if they trust Walter Cronkite
  when he delivers the nightly news
• Out of 1000, 570 say, they trust him
• 57 of the people trust Walter
• is 0.57
• If you collect another set of 1000 television
  viewers, what will the rating be?

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Confidence Statement

• We need to add a confidence statement
• We need to say something about the margin of
  error
• Confidence statements are based on the
  distribution of the values of the sample
  proportion that would occur if many
  independent SRS were taken from the same
  population
• The sampling distribution of the statistic

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Terminology Review

• Sample
• Population
• Statistic
• a numerical characteristic associated with a
  sample
• Parameter
• A numerical characteristic associated with the
  population
• Sampling error
• The need for interval estimation

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Point Estimation

• Point estimation of a parameter is the value of a
  statistic that is used to estimate the parameter
• Compute statistic (e.g., mean)
• Use it to estimate corresponding population
  parameter
• Point Estimators of Population Parameters(see
  next slide)

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Point Estimators for Population Parameters
Population Sample Calculating Paramete
r statistic formula
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Interval Estimation

• Sample point estimators are usually not
  absolutely precise
• How close or how distant is the calculated sample
  statistic from the population parameter
• We can say that the sample statistic is within a
  certain range or interval of the population
  parameter.
• The determination of this range is the basis for
  interval estimation

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Interval Estimation (2)

• A confidence interval (CI) represents the level
  of precision associated with a population
  estimate
• Width of the interval is determined by
• Sample size,
• variability of the population, and
• the probability level or the level of confidence
  selected

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Sampling Distributionof the Mean

• The distribution of all possible sample means for
  a sample of a given size
• Use the mean of a sample to estimate and draw
  conclusions about the mean of that entire
  population
• So we have samples of a particular size
• We need formulas to determine the mean and the
  standard deviation of all possible sample means
  for samples of a given size from a population

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Sample and Population Mean

• For samples of size n, mean of the variable
• Is equal to the mean of the variable under
  consideration
• Mean of all possible sample means is equal to the
  population mean

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Sample Standard Deviation

• For samples of size n, the standard deviation of
  the variable
• Is equal to the standard deviation of the
  variable under consideration, divided by the
  square root of the sample size
• For each sample size, the standard deviation of
  all possible sample means equals the population
  standard deviation divided by the square root of
  the sample size

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Central Limit Theorem

• Suppose all possible random samples of size n are
  drawn from an infinitely large, normally
  distributed population having a mean and a
  standard deviation
• The frequency distribution of these sample means
  will have
• A mean of (the population mean)
• A normal distribution around this population mean
• A standard deviation of

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Sampling Error

• Standard Error of the mean (SEM) is a basic
  measure for the amount of sampling error
• SEM indicates how much a typical sample mean is
  likely to differ from a true population mean
• Sample size, and population standard deviation
  affect the sampling error

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Sampling Error (2)

• The larger the sample size, the smaller the
  amount of sampling error
• The larger the standard deviation, the greater
  the amount of sampling error

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Finite Population Correction Factor

• The frequency distribution of the sample means is
  approximately normal if the sample size is large
• N lt 30 (small sample) N gt 30 (large sample)
• If you have a finite population, then you need to
  introduce a correction, i.e., the fpc rule/factor
  in the estimation process
• where fpc finite population correction
• n sample size
• N population size

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Standard Error of the Mean for Finite Populations

• When including the fpc should be
• In general, you include the fpc in the
  population estimates only when the ratio of
  sample size to population size exceeds 5 or
• when n / N gt 0.05

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Constructing Confidence Intervals

• A random sample of 50 commuters reveals that
  their average journey-to-work distance was 9.6
  miles
• A recent study has determined that the std.
  deviation of journey-to-work distance is
  approximately 3 miles
• What is the CI around this sample mean of 9.6
  that guarantees with 90 certainty that the true
  population mean is enclosed within that interval?

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Confidence Intervalfor the Mean

• Z value associated with a 90 confidence level
  (Z 1.65)
• The sample mean is the best estimate of the true
  population mean
• CI
• 9.6 1.65 (3/ ) 10.30 miles
• 9.6 - 1.65 (3/ ) 8.90 miles

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Confidence Interval

• We say that the sample statistic is within a
  certain range or interval of the population
  parameter
• e.g., in our sample, 57 of the viewers thought
  Walter Cronkite is trustworthy
• In the general population, between 54 and 60 of
  viewers think that Walter Cronkite is trustworthy
• Or, in our sample, the average commuting distance
  was 9.6 miles
• In the population, we calculated that the average
  commute is likely to be somewhere between 8.9
  miles and 10.3 miles

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Confidence Level

• Gives you an understanding of how reliable your
  previous statement regarding the confidence
  interval is
• The probability that the interval actually
  includes the population parameter
• For example, the confidence level refers to the
  probability that the interval (8.9 miles to 10.3
  miles) actually encompasses the TRUE population
  mean (90, 95, 99.7)
• Confidence Level probability is 1 - ?

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Significance Level

• ? (alpha)
• The probability that the interval that surrounds
  the sample statistic DOES NOT include the
  population parameter
• E.g., the probability that the average commuting
  distance does not fall between 8.9 miles and 10.3
  miles
• ? 0.10 (90) 0.05 (95) 0.01 (99.7)
• Confidence Interval width -- increases

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Sampling Error

• Total sampling error ?
• Probability that the sample statistic will fall
  into either tail of the distribution is
• ?/2
• If you want 99.7 confidence (i.e., low error),
  then you have to settle for giving a less precise
  estimate (the CI is wider)

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If the Standard Deviationis Unknown

• If we dont know the population mean, its likely
  we dont know the standard deviation
• What you are likely to have is the variance and
  standard deviation of your sample
• Also, you have a small population, so you have to
  use the finite population correction factor that
  was discussed earlier
• Once you have the formula for standard error,
  then you can proceed as before to determine the
  confidence interval

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Standard Error
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Students T Distribution

• William Gosset (1876-1937)
• Published his contributions to statistical theory
  under a pseudonym
• Students t distribution is used in performing
  inferences for a population mean, when,
• The population being sampled is approximately
  normally distributed
• The population standard deviation is unknown
• And the sample size is small (n lt 30)

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Characteristics of the t - Distribution

• A t curve is symmetric, bell shaped
• Exact shape of distribution varies with sample
  size
• When n nears 30, the value of t approaches the
  standard normal Z value
• A particular distribution is identified by
  defining its degrees of freedom (df)
• For a t distribution, df (n -1)

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Properties of t Curves

• The total area under a t curve 1
• A t curve extends indefinitely in both
  directions, approaching, but never touching the
  horizontal axis
• A t-curve is symmetrical about 0
• As the degrees of freedom become larger, t
  curves look increasingly like the standard normal
  curve
• We need to use a t-table and look for values of
  t, instead of Z to determine the confidence
  interval

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Calculating various CIs

• Sampling
• SRS, systematic, or stratified
• Parameters
• Mean, total, or proportion
• Six situations
• Consider whether to use fpc
• when n/N gt 0.05
• Consider whether to use Z or t
• when n lt 30

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If Random or Systematic Sample

• Estimate of Population Mean
• Best estimate is ?
• Estimate of sampling error
• Standard error of the mean (inc. fpc)

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If Stratified Sample

• Estimate of population mean
• Still equal to sample mean but
• Std. Error of the mean (inc. fpc)

Where mnumber of strata i refers to a
particular stratum
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Minimum Sample Size

• Before going out to the field, you want to know
  how big the sample ought to be for your research
  problem
• Sample must be large enough to achieve precision
  and CI width that you desire
• Formulas to determine the three basic population
  parameters with random sampling

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Sample Size Selection - Mean

• Your goal is to determine the minimum sample size
• You want to situate the estimated population
  mean, in a specified CI

E amount of error you are willing to tolerate
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Example 1

• We are looking at Neighborhood X
• 3,500 households
• Sample size 25 households
• Sample mean 2.73
• Sample variance 2.6
• CI 90
• Find the mean number of people per household

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Example 2

• Sample of 30 households
• Sample standard deviation is 1.25
• What sample size is needed to estimate the mean
  number of persons per household in neighborhood X
  
• and be 90 confident that your estimate will be
  within 0.3 persons of the true population mean?