Estimation

1
Estimation

• Goal Use sample data to make predictions
  regarding unknown population parameters
• Point Estimate - Single value that is best guess
  of true parameter based on sample
• Interval Estimate - Range of values that we can
  be confident contains the true parameter

2
Point Estimate

• Point Estimator - Statistic computed from a
  sample that predicts the value of the unknown
  parameter
• Unbiased Estimator - A statistic that has a
  sampling distribution with mean equal to the true
  parameter
• Efficient Estimator - A statistic that has a
  sampling distribution with smaller standard error
  than other competing statistics

3
Point Estimators

• Sample mean is the most common unbiased estimator
  for the population mean m

• Sample standard deviation is the most common
  estimator for s (s2 is unbiased for s2)

• Sample proportion of individuals with a (nominal)
  characteristic is estimator for population
  proportion

4
Confidence Interval for the Mean

• Confidence Interval - Range of values computed
  from sample information that we can be confident
  contains the true parameter
• Confidence Coefficient - The probability that an
  interval computed from a sample contains the true
  unknown parameter (.90,.95,.99 are typical
  values)
• Central Limit Theorem - Sampling distributions of
  sample mean is approximately normal in large
  samples

5
Confidence Interval for the Mean

• In large samples, the sample mean is
  approximately normal with mean m and standard
  error
• Thus, we have the following probability statement

• That is, we can be very confident that the
  sample mean lies within 1.96 standard errors of
  the (unknown) population mean

6
Confidence Interval for the Mean

• Problem The standard error is unknown (s is also
  a parameter). It is estimated by replacing s with
  its estimate from the sample data

95 Confidence Interval for m
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Confidence Interval for the Mean

• Most reported confidence intervals are 95
• By increasing confidence coefficient, width of
  interval must increase
• Rule for (1-a)100 confidence interval

8
Properties of the CI for a Mean

• Confidence level refers to the fraction of time
  that CIs would contain the true parameter if
  many random samples were taken from the same
  population
• The width of a CI increases as the confidence
  level increases
• The width of a CI decreases as the sample size
  increases
• CI provides us a credible set of possible values
  of m with a small risk of error

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Confidence Interval for a Proportion

• Population Proportion - Fraction of a population
  that has a particular characteristic (falling in
  a category)
• Sample Proportion - Fraction of a sample that has
  a particular characteristic (falling in a
  category)
• Sampling distribution of sample proportion (large
  samples) is approximately normal

10
Confidence Interval for a Proportion

• Parameter p (a value between 0 and 1, not
  3.14...)
• Sample - n items sampled, X is the number that
  possess the characteristic (fall in the category)
• Sample Proportion
• Mean of sampling distribution p
• Standard error (actual and estimated)

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Confidence Interval for a Proportion

• Criteria for large samples
• 0.30 lt p lt 0.70 ?? n gt 30
• Otherwise, X gt 10, n-X gt 10
• Large Sample (1-a)100 CI for p

12
Choosing the Sample Size

• Bound on error (aka Margin of error) - For a
  given confidence level (1-a), we can be this
  confident that the difference between the sample
  estimate and the population parameter is less
  than za/2 standard errors in absolute value
• Researchers choose sample sizes such that the
  bound on error is small enough to provide
  worthwhile inferences

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Choosing the Sample Size

• Step 1 - Determine Parameter of interest (Mean or
  Proportion)
• Step 2 - Select an upper bound for the margin of
  error (B) and a confidence level (1-a)

Proportions (can be safe and set p0.5)
Means (need an estimate of s)
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Confidence Interval for Median

• Population Median - 50th-percentile (Half the
  population falls above and below median). Not
  equal to mean if underlying distribution is not
  symmetric
• Procedure
• Sample n items
• Order them from smallest to largest
• Compute the following interval
• Choose the data values with the ranks
  corresponding to the lower and upper bounds