Inferences About the Mean-I (Large Samples)

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Inferences About the Mean-I(Large Samples)

• QSCI 381 Lecture 21
• (Larson and Farber, Sect 6.1)

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Statistical Inference

• Sample statistics can be used to
• the value(s) of unknown
• population parameter(s). We are making inference
  about the unknown population parameter(s) based
  on data.
• We start by making estimates of the population
  parameter ?
• when the sample size is large (?30)
• when the sample size is small.

estimate
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Point Estimation

• A is a single value
  for a population parameter.
• The most unbiased point estimate of the
  population mean ? is the sample mean .
• Warning is an estimate and not the population
  mean the difference between the two is the
  uncertainty of the estimate.

point estimate
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Example
An acoustic survey involved three passes of an
area. The average density during the first pass
was 52.3.
Results for passes 2 and 3
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Interval Estimation

• An is an interval,
  or range of values, used to estimate a population
  parameter.

interval estimate
Point estimate52.3 Interval (48.5, 58.1)
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Level of Confidence

• The is the
  probability that the interval estimate contains
  the population parameter.

level of confidence, c
c
For n?30, the sampling distribution of sample
means is a normal distribution. The level of
confidence, c, is the area under the standard
normal distribution between zc and zc. Hint
for zc1.645, there is a 90 probability that
the interval estimate contains the population
mean. Why do I say this?
½(1-c)
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Extent of Error

• Given a level of confidence, c, the
• 
  (margin of error or error tolerance) is the
  greatest possible difference between the point
  estimate and the value of the parameter being
  estimated

maximum error of estimate
E
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Example-I

• Use the data for the first pass of the acoustic
  survey and a 95 level of confidence (i.e.
  zc1.96) to find the maximum error of estimate
  for the mean density. The steps to calculate the
  maximum error of estimate are
• Find the sample statistics n and .
• Specify ? if known. Otherwise, if n?30, find the
  sample standard deviation, s, and use this as an
  estimate of ?.
• Find the level of z that corresponds to the
  confidence level.
• Find the maximum error of estimate of E.

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Example-II

• Application of these steps gives
• n50
• ? is not known so we estimate it from the sample
• The critical value of z is 1.96.
• The maximum error of estimate E is
• We are 95 confident that the maximum error of
  estimate for the population mean is about 2.95
  units of density.

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Confidence Intervals for the Population Mean
c-confidence interval

• A for the
  population mean ? is
• The probability that the confidence interval
  contains ? is c.
• The 95 confidence interval for the mean density
  is (52.28-2.95, 52.382.95) (49.3, 55.2)

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Confidence Intervals for the Population Mean

• To compute a (95) confidence interval using
  EXCEL
• AVERAGE(D2D51)STDEV(D2D51)/SQRT(COUNT(D2D51))
  NORMINV(0.025,0,1)
• AVERAGE(D2D51)STDEV(D2D51)/SQRT(COUNT(D2D51))
  NORMINV(0.975,0,1)
• Note The calculation is based on STDEV and not
  STDEVP.

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Example-III(? known to be 10)

• Application of the steps gives
• n50 ? 10
• The critical value of z is 1.96.
• The maximum error of estimate E is
• We are 95 confident that the maximum error of
  estimate for the population mean is about 2.77
  units of density.

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Interpretation
100 simulated confidence intervals for the mean
of acoustic survey density
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Summary
Given c find zc
Compute
Is ? known?
No
Yes