Estimation in Sampling

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Estimation in Sampling

• How many cases do I need?
• How good is my sample?
• References Julio Rivera

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Stop for a Moment

• What is our goal in sampling?
• We want to make an inference about something
  using as little information as prudently possible
• Sometimes we want to be more certain than others
• How do you guarantee accuracy plus or minus 3?

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Stop for a Moment

• How do you know how many observations to make?
• How do you know how many points to sample?
• How do you know how many people to question?

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Two types of Estimation

• Point and Interval Estimation

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Point Estimation

• Simple concept
• Statistic is calculated from a sample
• Then you estimate the population parameter
• For a point estimate of the mean use a sample
  mean
• For a point estimate of the standard deviation,
  use a sample standard deviation

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• In your book note that the formula for the
  standard deviation uses n-1 in the denominator.
• For samples less than 30 use n-1. For samples
  greater than 30 use n.
• For small samples the n-1 formula better
  represents the standard deviation

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How good are point estimates

• Always some error.
• Unlikely that the sample statistic is going to
  equal the population parameter
• But we want to know how good this sample
  statistic is
• Therefore we construct a confidence interval

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Interval Estimates

• An interval bounded by two values used to
  estimate the value of a population parameter.
• The values that bound this interval are
  calculated from the sample that is being used as
  the basis for the estimation

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Level of Confidence

• 1-a (a probability)
• The probability that the sample selected yields
  boundary values that lie on opposites sides of
  the parameter being estimated
• For instance
• The probability the mean of the sample is between
  the two boundaries

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Confidence Interval

• An interval estimate with a specified level of
  confidence
• In other words it is the level of precision
  associated with the sample estimate
• Determined by sample size, variability of the
  sample and the confidence selected
• But before we look at CIs more closely we need to
  go back and look at the Central Limit Theorem

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Central Limit Theorem

• If all possible random samples, each of size n,
  are taken from any population with a mean m and a
  standard deviation s, the sampling distribution
  of sample means will
• Have a mean of m
• have a standard deviation of s of s/Ön
• Be normally distributed or approximately normal
  for samples of 30 or more when the parent sample
  is normal

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Qualities of the Central Limit Theorem

• In essence this is the mean of the means
• The larger the sample--the closer the sample mean
  will be to the population mean the smaller the
  amount of sampling error
• The larger the standard deviation, the larger the
  amount of sampling error

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The standard error of the mean

• is the standard deviation of these sample means
• Divide the population standard deviation by the
  square root of the sample size

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Why are these valuable?

• We often deal with populations which are not
  normally distributed
• However, the collection of means of the samples
  of those populations are normally distributed
  (n30)

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The Central Limit Theorem allows us to put a
confidence interval around our sample mean

• It allows us to decide how good our sample is

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Confidence Intervals and Estimation

• You want to know that your sample mean falls with
  in a certain range with a level of confidence
• You are trying to estimate the population mean
  using your sample mean
• A confidence interval is constructed by
  establishing the interval (a z-score
  corresponding to the CI) from the mean.

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Confidence Interval

• If you wanted a 95 CI from a mean of 35.6.
• 95z score of 1.96 (Table A)
• If we think the population standard deviation
  is16
• Our n100

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Confidence Interval

• What we have suggested here is that the
  probability is that 95 the true sample mean is
  between 32.46 and 38.74

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Confidence Interval

• In a 95 CI, the probability of being wrong is 5
  (a)
• This alpha error is divided in half and
  distributed out on the tails of the normal
  distribution (a/2)
• The case at right is from you book and uses a 90
  interval

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Some things to note

• If you dont know the population standard
  deviation, use the sample standard deviation
• Likewise, if you don't know the population
  variance, use the sample variance

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Being Rigorous

• Generally you shouldnt set a confidence interval
  below .95
• Allows a precise estimate of the population
  parameter
• If you can, set it at .99 (Often this is too
  expensive and gives too wide an interval)
• However there are some types of data which you
  may be able to set lower CIs
• Do this on a case by case basis

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Using t instead of z

• For sample sizes less than 30 use the students t
  value rather than z
• Distribution is symmetric and bell
  shaped--approaches normal near n30

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Middletown Example

• Planners are asking several questions because
  they believe the census data is out of date
• What is the mean number of people in each
  household?
• How many people are in the community?
• The proportion of households with a child under
  18 years of age

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First step

• How many people per household
• Small random sample
• Planners sample 25 households
• Set a 90 confidence level
• Important values are at left

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• Ignore the finite population correction
• Resulting interval is a little large
• The planners decide to increase the sample size
  to 250
• The decide to keep the level of confidence .90

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• The new mean is 2.68
• variance is 4.3
• n250
• we use z rather than t (1.65)
• Finite population correction factor is used
• Changes narrow the CI

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How Many People are in Middletown?

• Make an estimate by multiplying the number of
  households by the mean number of persons you just
  calculated

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How good is this estimate?

• Lets construct the confidence interval
• Use the formulas for random and systematic
  samples from your book
• 90 probability that the true mean is between
  8650.16 and 10,109.84

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Children

• How many households have children
• of the 250 households surveyed--105 have children
• Our best estimate of proportion is .42
• Once again, we want to construct a CI--use
  formulas for random and systematic samples

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• Plug in the values
• The probability that true mean of the proportion
  of households with families is between 0.37 and
  0.47 is 90
• Translates out to numbers

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How large should my n be?

• There are a number of methods for determining
  what size n you should do for your research
• We will go over a few of the more standard ways
• This information is important because you can
  save yourself time and money if you know how to
  predict the sample size you need

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Think before you sample

• What kind of sample
• What population parameter are you estimating
  (mean, total, proportion)
• What level of precision (width of your confidence
  interval that can be tolerated)
• Level of confidence to be obtained from sample
• Types of statistical tests you will use (more
  important later)

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Determining Sample Size

• Use these formulas when you want to determine
  sample size for a confidence interval around a
  mean

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Determining Sample Size

• A sample of what size would be needed to estimate
  a population mean within six units with 98
  confidence if the population has a standard
  deviation of 16?

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Determining Sample Size

• Plug in the values
• Round up to 39
• Most of the time we dont know the standard
  deviation
• Do a pilot of n30
• Use this and then take additional sample units
• This is two-stage sampling

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• Suppose the total you want to estimate the total
  population of Middletown within 1000 persons
• How many households would you need to sample
• N3500 z1.65 s2.05

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Sample Size using proportion

• A pilot study suggests that 36 of households
  contain one or more child
• We want to know how many households to sample

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A Political Poll

• The Rivera Opinion Poll Company wishes to
  determine the percentage of eligible voters who
  would approve the levying of additional property
  taxes to finance public schools in Milwaukee
• The probability should be about 99.7 that the
  percentage estimated would be plus or minus 3.
• How large a sample do we need?

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• The most conservative assumption of p and p-1 is
  .50 for each (its a mathematical property)
• We need to sample 2500 people for this level of
  accuracy

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• If you keep plus minus 3
• And change the probability (z) to 95

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Estimation in Sampling

• What a nice thing to know how to make these
  estimates
• They will save you (if you do graduate research)
  time and money
• They will save your employer time and money (this
  will make you look good)