Estimation of Means and Proportions

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Estimation of Means and Proportions
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Concepts

• Estimator a rule that tells us how to estimate
  a value for a population parameter using sample
  data
• Estimate a specific value of an estimator for
  particular sample data

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Concepts

• A point estimator is a rule that tells us how to
  calculate a particular number from sample data to
  estimate a population parameter
• An interval estimator is a rule that tells us how
  to calculate two numbers based on sample data,
  forming a confidence interval within which the
  parameter is expected to lie

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Properties of a Good Estimator

• Unbiasedness mean of the sampling distribution
  of the estimator equals the true value of the
  parameter
• Efficiency The most efficient estimator among a
  group of unbiased estimators is the one with the
  smallest variance

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Properties of a Good Estimator
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Estimation of a Population Mean

• The CLT suggests that the sample mean may be a
  good estimator for the population mean. The CLT
  says that
• Sampling distribution of sample mean will be
  approximately normally distributed regardless of
  the distribution of the sampled population if n
  is large
• The sample mean is an unbiased estimator
• The standard error of the sample mean is

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Estimation of a Population Mean

• A point estimator of the population mean is
• An interval estimator of the population mean is a
• confidence interval,
  meaning that the true population parameter lies
  within the interval
  
• of the time, where
  is the z value corresponding to an area
  in the upper tail of a standard normal
  distribution

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Estimation of a Population Mean

• Usually s (the population standard deviation) is
  unknown.
• If n is large enough (n 30) then we can
  approximate it with the sample standard deviation
  s.

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One Sided Confidence Intervals

• In some cases we may be interested in the
  probability the population parameter falls above
  or below a certain value
• Lower One Sided Confidence Interval (LCL)
• LCL (point estimate)
• Upper One Sided Confidence Interval (UCL)
• UCL (point estimate)

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Small Sample Estimation of a Population Mean

• If n is large, we can use sample standard
  deviation s as reliable estimator of population
  standard deviation
• No matter what distribution the population has,
  sampling distribution of sample mean is normally
  distributed
• As the sample size n decreases, the sample
  standard deviation s becomes a less reliable
  estimator of the population standard deviation
  (because we are using less information from the
  underlying distribution to compute s)
• How do we deal with this issue?

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t Distribution

• Assume
• (1) The underlying population is normally
  distributed
• (2) Sample is small and s is unknown
• Using the sample standard deviation s to replace
  s, the t statistic
• follows the t distribution

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Properties of the t Distribution

• mound-shaped
• perfectly symmetric about t0
• more variable than z (the standard normal
  distribution)

• affected by the sample size n (as n increases s
  becomes a better approximation for s)
• n-1 is the degrees of freedom (d.f.) associated
  with the t statistic

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More on the t Distribution

• Remember the t-distribution is based on the
  assumption that the sampled population possesses
  a normal probability distribution.
• This is a very restrictive assumption.
• Fortunately, it can be shown that for non-normal
  but mound-shaped distributions, the distribution
  of the t statistic is nearly the same shape as
  the theoretical t-distribution for a normal
  distribution.
• Therefore the t distribution is still useful for
  small sample estimation of a population mean even
  if the underlying distribution of x is not known
  to be normal

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How to use the t-distribution table

• The t-distribution table is in the book
  (Appendix II, Table 4, pp611). ta is the value
  of t such that an area a lies to its right.
• To use the table
• Determine the degrees of freedom
• Determine the appropriate value of a Lookup the
  value for ta

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Table t Distribution
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The Difference Between Two Means

• Suppose independent samples of n1 and n2
  observations have been selected from populations
  with means , and variances ,
  
• The Sampling Distribution of the difference in
  means ( ) will have the following
  properties

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The Difference Between Two Means

• The mean and standard deviation of
  is
• If the sampled populations are normally
  distributed, the sampling distribution of (
  ) is exactly normally distributed regardless
  of n
• If the sampled populations are not normally
  distributed, the sampling distribution of (
  ) is approximately normally distributed when
  n1 and n2 are large

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Point Estimation of the Difference Between Two
Means

• Point Estimator
• A confidence interval for (
  ) is

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Difference Between Two Means (small sample)

• If n1 and n2 are small then the t statistic
• is distributed according to the t distribution
  if the following assumptions are satisfied
• 1. Both samples are drawn from populations
  with a normal distribution
• 2. Both populations have equal variances

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Difference Between Two Means (small sample)

• In practice, the t statistic is still appropriate
  even if the underlying distributions are not
  exactly normally distributed.
• To compute s, we can pool the information from
  both samples
• or

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Difference Between Two Means (small sample)

• Point Estimate
• Interval Estimate
• a confidence interval
  for
• is
• Where s is computed using the pooled estimate
  described earlier

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Sampling Distribution of Sample Proportions

• Recall from Chapter 6
• If a random sample of n objects is selected from
  the population and if x of these possess a
  chararacteristic of interest, the sample
  proportion is
• The sampling distribution of will have a
  mean and standard deviation

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Estimators for p

• Assuming n is sufficiently large and the interval
  lies in the interval from 0 to 1, the
• Point Estimator for p
• Interval Estimator for p
• A confidence interval for p
  is

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Estimating the Difference Between Two Binomial
Proportions

• Point estimate
• Confidence interval for the difference

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Choosing Sample Size

• How many measurements should be included in the
  sample?
• Increasing n increases the precision of the
  estimate, but increasing n is costly
• Answer depends on
• What level of confidence do you want to have
  (i.e., the value of 100(1- a )?
• What is the maximum difference (B) you want to
  permit between the estimate of the population
  parameter and the true population parameter

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Choosing Sample Size

• Once you have chosen B and a, you can solve the
  following equation for sample size n
• If the resulting value of n is less than 30 and
  an estimate

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Choosing Sample Size