LSSG Black Belt Training

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LSSG Black Belt Training
Estimation Central Limit Theorem and Confidence
Intervals
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Central Limit Theorem

• Assume a population with a non-normal
  distribution.

Mean µ Stdev s
If we took a sample of size 50 from this
population, what would it look like?
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CLT - Multiple Samples from the same Population

• Each sample of n50 from the same population will
  tend to look like the population, and the sample
  means will be close to the population mean.
• The sample means are unbiased estimators of the
  population mean. They will vary randomly above
  and below the actual population mean.
• If all such samples (n50) were drawn, how would
  the sample means be distributed?

X1
X2
X3
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CLT Sampling Distribution of X

• Most sample means will be close to the population
  mean.
• Some sample means will be a little farther away.
• A few will be quite a bit off the mark.
• A rare number will be extremely far away.
• In other words, the X values will be
  approximately normally distributed.

How would the mean of this distribution compare
to the original population mean? How about the
standard deviation of this distribution? How
would sample size affect this relationship?
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Central Limit Theorem Statement

• For sufficiently large sample sizes (typically
  ngt30), the distribution of the sample means
  (X-Bar) is approximately normal, and
• Mean of sample means Population Mean
• Standard Deviation of sample means
• (Std Dev. of Population/ square root of n)
• This standard deviation of the sample means is
  also called the standard error.
• Additional inference
• Since the X-bars are normally distributed, 95 of
  all samples (large enough n) from a population
  will yield an X-bar that is within 2 standard
  errors from the population mean.

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Confidence Intervals

• We take a sample of 64 parts from a population,
  and want to estimate the population mean of the
  part length. The sample mean is 25 mm. The
  population standard deviation is known to be 0.2
  mm.
• From CLT, we know that this sample mean (25) is
  within 2 standard errors (actually 1.96) of the
  population mean, with 95 confidence.
• Hence the reverse is also true.
• Thus, population mean is
• X-bar 2 SE
• Here, SE 0.16 / v64 0.16/8 0.02
• Thus 95 CI for µ is given by
• 25 20.02, or 25 0.04 mm
• The value 0.04 is the Margin of Error (MOE)

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Confidence Intervals Unknown s

• In reality, s is generally unknown, and must be
  substituted with s, the sample standard
  deviation. In that case, the margin of error is
  higher, and is computed using the t-distribution
  rather than the standard normal (z dist). Thus
  instead of 1.96 standard errors for 95
  confidence, we use a larger number obtained from
  the t-tables.
• (In Excel, type tinv(0.05,df), where df is the
  degrees of freedom, equal to n-1. Here in the
  previous problem, df is 63).