Statistical Intervals Based on a Single Sample

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Lecture 14

• Chapter 7
• Statistical Intervals Based on a Single Sample

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• What is a confidence interval (CI)?
• The point estimate provides only a single value
  estimate for a population parameter. It does not
  provide any information on how good the
  estimate is, or how close it is to the real value
  of the parameter.
• A confidence interval provides an interval along
  with a certain confidence probability value (1??)
  for a population parameter. A CI indicates the
  percentage of the CIs formed from a number of
  different samples of same size that would contain
  the real value of the estimated population
  parameter.
• For example
• Let the the 95 confidence interval for the mean
  product length, calculated using a certain
  sample, is 24,29. This means that if we kept
  taking similar samples to which we calculated the
  above CI from, about 95 of the CIs that we form
  will contain the real value of the mean length.
  Also, since (1??) 0.95, ? 0.05

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• Types of CIs
• There are three types of CIs
• Two-sided CI ?L ? ? ? ?U
• Lower one-sided CI ?L ? ? ? ?
• Upper one-sided CI -? ? ? ? ?U
• There are three types of continuous random
  variables.
• Nominal-the-best (NTB) ? ?L ? ? ? ?U
• Larger-the-best (LTB) ? ?L ? ? ? ?
• Smaller-the-best (STB) ? -? ? ? ? ?U
• Most common CIs are 95, 99, and 90 CIs.

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• Some Examples
• Nominal-the-better
• Clearance, chemical content (pH level), etc.
• We would like the value to be between two
  comsumer specification limits (LSL m - ? and
  USL m ?).
• Larger-the-best
• Strength, lifetime, reliability, etc.
• We would like the value to be larger than a
  consumer lower specification limit LSL only.
• Smaller-the-better
• Amount of error, time delay, monetary loss, etc.
• We would like the value to be smaller than a
  consumer upper specification limit USL only.

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• In general, to develop a parametric CI for a
  parameter ?, the sampling distribution (SMD) of
  its point estimator must be known and then used
  appropriately.
• Further, to obtain a lower one-sided CI, the
  value of ? should generally be placed at the
  upper tail of a distribution and vice a versa for
  an upper one-sided CI.

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• CONFIDENCE INTERVAL FOR THE MEAN OF A NORMAL
  PROCESS
• WHEN THE VARIANCE IS KNOWN

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CONFIDENCE INTERVAL FOR THE MEAN OF A NORMAL
PROCESS WHEN THE VARIANCE IS UNKNOWN

• In the previous example, the population variance
  was assumed to be known. However, in reality it
  is very likely that the real value of the
  variance will not be known.
• As the central limit theorem states, the sample
  average has approximately a normal distribution,
  whatever the parent distribution is, and

• However, since the value of ? is not known, we
  will use the unbiased estimator S instead of ?.
  The standardized variable

• follows approximately the standard normal
  distribution only if n gt 40. Therefore, we will
  use the standard normal distribution only if n gt
  40, but not if n lt 40.

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• For small sample sizes, or if nlt40
• The standardized variable will be denoted by T

• follows a t (Students t) distribution with
  (n-1) degrees of freedom (df). The df is denoted
  by ?.
• The t distribution is always more spread out than
  the standard normal distribution, which accounts
  for the added variability due to the uncertainty
  about the real value of the population variance.
• For n gt 40, the t distribution becomes
  practically equal to the standard normal
  distribution.

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• Confidence Interval for the Variance of a Normal
  Universe

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• Last topic Tolerance Intervals