Estimation

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Chapter 8

• Estimation

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I. Introduction

• A. Estimation the process of inferring the
  values of unknown population parameters from
  known sample statistics

• B. Estimator the type of sample statistic that
  is used to make inferences about a given type of
  population parameter

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• II. Choosing a Good Estimator

There are three major criteria that are employed
when one considers the question of what makes a
good estimator

• A. Unbiased Estimator if the mean of all
  possible values of that statistic equals the
  population parameter the statistic seeks to
  estimate.

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II. Choosing a Good Estimator
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II. Choosing a Good Estimator

• B. Efficient Estimator among all the available
  unbiased estimators, the sample statistic that
  has the smallest variance for a given sample size

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II. Choosing a Good Estimator
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II. Choosing a Good Estimator

• C. Consistent Estimator a sample statistic
  whose value gets closer to the parameter being
  estimated as sample size increases

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II. Choosing a Good Estimator
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II. Choosing a Good Estimator

• D. Mean Squared Error the sum of an estimators
  squared bias plus its variance, or (MuE T)2
  VarianceE
• In terms of Figure 8.2, this procedure picks the
  statistic with sampling distribution B as the
  best estimator.

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II. Choosing a Good Estimator
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• III. Types of Estimates

• A. Point Estimate when the estimate of a
  population parameter is expressed as a single
  numerical value

• B. Interval Estimate a range of values within
  which the unknown population parameter presumably
  lies

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• IV. Making Point Estimates

Whenever the statistical population of interest
is normally distributed or when-ever the
conditions of the Central Limit Theorem are
fulfilled (n gt 30, but n lt 0.05N). And
regardless of whether sampling occurs from a
large or a small population, the sample mean is
an unbiased, efficient, and consistent estimator
of the population mean.
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Problem 8.2 - The Length of a Manuscript
A publisher wishes to estimate the number of
words in a 935-page manuscript. A random sample
of 30 pages is taken. Because nothing is known
about the distribution of the words-per-page
population, but n gt 30 as well as n lt 0.05N, the
Central Limit Theorem can be invoked.
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Problem 8.2 - The Length of a Manuscript
Thus, Mu x-bar Mu, and the sample mean can be
used as an unbiased estimator of Mu. As the
remainder of Table 8.2 shows, the sample mean is
248.4 words per page this sample mean becomes
the point estimate of the population mean,
Mu. Hence, the length of the manuscript is
estimated at 248.4(935) 232,254 words.
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Problem 8.2 - The Length of a Manuscript
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Problem 8.2 - The Length of a Manuscript
The standard deviation of words per page for the
entire manuscript can be estimated also. The
sample variance, s2, has been calculated by means
of the shortcut method in Box 3.M it becomes the
point estimate of the population variance. Thus,
the population standard deviation can be
estimated at square root of 747.49 27.34 words.
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• V. Making Interval Estimates

Even though an unbiased point estimator will on
the average take on a value equal to the
parameter being estimated, any one estimate is
unlikely to be on target. Yet, typically, only
one estimate is made because only one sample is
taken (given the constraints). Hence, the
typical point estimate is almost certain to lie
below or above the true value of the parameter.
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V. Making Interval Estimates

• A. Constructing the Interval

Statisticians routinely construct intervals by
making the point estimate the interval center and
creating a range below and above the center with
the help of the estimators standard error.
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V. Making Interval Estimates

• B. Confidence Interval
• If simple random samples of size n are repeatedly
  taken from a given population, many different
  values of a given sample statistic will be found.
  Any one sample statistic, therefore, may well
  produce an incorrect estimate of an unknown
  population parameter. However, by subtracting
  and adding a certain amount, a sample statistic
  can be turned into a range of values among which
  the unknown population parameter can presumably
  be found.

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V. Making Interval Estimates

• C. The Nature of Confidence Intervals
• Given sample size, n, the level of confidence
  attached to an estimate varies directly with the
  value of z and, thus, with the interval width.
  Notice how a smaller z value means greater
  precision of an estimate (a narrower interval)
  but also implies a smaller degree of confidence
  in the estimate. A larger z value means less
  precision (a wider interval) but implies a
  greater degree of confidence.

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V. Making Interval Estimates

• D. The Meaning of Confidence Level the
  percentage of intervals that can be expected to
  contain the parameters actual value when the
  same procedure of interval construction is used
  again and again

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V. Making Interval Estimates
CAUTION
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• E. A Graphical Illustration

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• E. A Graphical Illustration

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• VI. Making Interval Estimates of the Population
  Mean Large Samples

• Here we will focus on
• large samples (n gt 30) and
• normally distributed sampling distributions

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VI. Making Interval Estimates of the Population
Mean Large Samples
EXAMPLE PROBLEM 8.5
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VI. Making Interval Estimates of the Population
Mean Large Samples
SOLUTION
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VI. Making Interval Estimates of the Population
Mean Large Samples
EXAMPLE PROBLEM 8.6
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VI. Making Interval Estimates of the Population
Mean Large Samples
SOLUTION
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• VII. Making Interval Estimates of the Population
  Mean Small Samples

• Here we will focus on
• small samples (n lt 30)
• (2) normally distributed sampling
• distributions
• (3) absence of any other information about the
  population

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VII. Making Interval Estimates of the Population
Mean Small Samples

• A. Student's t distribution
• single-peaked above the random variables mean,
  median, and mode of zero,
• perfectly symmetrical about this central value,
  and
• characterized by tails extending indefinitely in
  both directions from the center, approaching but
  never touching the horizontal axis.

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VII. Making Interval Estimates of the Population
Mean Small Samples
A. Student's t distribution
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VII. Making Interval Estimates of the Population
Mean Small Samples

• B. Degrees of Freedom (n 1 )
• Equal the number of values that can be freely
  chosen when calculating a statistic.

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VII. Making Interval Estimates of the Population
Mean Small Samples
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VII. Making Interval Estimates of the Population
Mean Small Samples
Upper and lower limits of confidence interval for
the population mean (using the t distribution)
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VII. Making Interval Estimates of the Population
Mean Small Samples
EXAMPLE PROBLEM 8.10
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VII. Making Interval Estimates of the Population
Mean Small Samples
SOLUTION
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VII. Making Interval Estimates of the Population
Mean Small Samples
CAUTION