5:%20Introduction%20to%20estimation

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5 Introduction to estimation

1. Intro to statistical inference
2. Sampling distribution of the mean
3. Confidence intervals (s known)
4. Students t distributions
5. Confidence intervals (s not known)
6. Sample size requirements

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Statistical inference

• Statistical inference ? generalizing from a
  sample to a population with calculated degree of
  certainty
• Two forms of statistical inference
• Estimation ? introduced this chapter
• Hypothesis testing ? next chapter

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Parameters and estimates

• Parameter ? numerical characteristic of a
  population
• Statistics a value calculated in a sample
• Estimate ? a statistic that guesstimates a
  parameter
• Example sample mean x-bar is the estimator of
  population mean µ

Parameters and estimates are related but are not
the same
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Parameters and statistics
Parameters Statistics
Source Population Sample
Notation Greek (µ, s) Roman (x, s)
Random variable? No Yes
Calculated No Yes
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Sampling distribution of the mean

• x-bar takes on different values with repeated
  (different) samples
• µ remain constant
• Even though x-bar is variable, its behavior is
  predictable
• The behavior of x-bar is predicted by its
  sampling distribution, the Sampling Distribution
  of the Mean (SDM)

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Simulation experiment

• Distribution of AGE in population.sav (Fig.
  right)
• N 600
• µ 29.5 (center)
• s 13.6 (spread)
• Not Normal (shape)
• Conduct three sampling simulations
• For each experiment
• Take multiple samples of size n
• Calculate means
• Plot means ? simulated SDMs
• Experiment A each sample n 1
• Experiment B each sample n 10
• Experiment C each sample n 30

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Results of simulation experiment

• Findings
• SDMs are centered on 29 (µ)
• SDMs become tighter as n increases
• SDMs become Normal as the n increases

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95 Confidence Interval for µ
Formula for a 95 confidence interval for µ when
s is known
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Illustrative example

• Example
• Population with s 13.586 (known ahead of time)
• SRS ? 21, 42, 11, 30, 50, 28, 27, 24, 52
• n 10, x-bar 29.0
• SEM s / ?n 13.586 / ?10 4.30
• 95 CI for µ
• xbar (1.96)(SEM)
• 29.0 (1.96)(4.30)
• 29.0 8.4
• (20.6, 37.4)

Margin of error
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Margin of error

• Margin or error ? d half the confidence
  interval
• Surrounded x-bar with margin of error
• 95 CI for µ
• xbar (1.96)(SEM)
• 29.0 (1.96)(4.30)
• 29.0 8.4

point estimate
margin of error
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Interpretation of a 95 CI
We are 95 confident the parameter will be
captured by the interval.
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Other levels of confidence
Let a ? the probability confidence interval will
not capture parameter 1 a ? the confidence level
Confidence level 1 a Alpha level a z1a/2
.90 .10 1.645
.95 .05 1.96
.99 .01 2.58
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(1 a)100 confidence for µ
Formula for a (1-a)100 confidence interval for µ
when s is known
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Example 99 CI, same data

• Same data as before
• 99 confidence interval for µ
• x-bar (z1.01/2)(SEM)
• x-bar (z.995)(SEM)
• 29.0 (2.58)(4.30)
• 29.0 11.1
• (17.9, 40.1)

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Confidence level and CI length
p. 5.9 demonstrates the effect of raising your
confidence level ? CI length increases ? more
likely to capture µ
Confidence level CI for illustrative data CI length
90 (21.9, 36.1) 14.2
95 (20.6, 37.4) 16.8
99 (17.9, 40.1) 22.2
CI length UCL LCL
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Beware

• Prior CI formula applies only to
• SRS
• Normal SDMs
• s known ahead of time
• It does not account for
• GIGO
• Poor quality samples (e.g., due to non-response)

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When s is Not Known

• In practice we rarely know s
• Instead, we calculate s and use this as an
  estimate of s
• This adds another element of uncertainty to the
  inference
• A modification of z procedures called Students t
  distribution is needed to account for this
  additional uncertainty

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Students t distributions
Brilliant!

• William Sealy Gosset (1876-1937) worked for the
  Guinness brewing company and was not allowed to
  publish
• In 1908, writing under the the pseudonym
  Student he described a distribution that
  accounted for the extra variability introduced by
  using s as an estimate of s

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t Distributions

• Students t distributions are like a Standard
  Normal distribution but have broader tails
• There is more than one t distribution (a family)
• Each t has a different degrees of freedom (df)
• As df increases, t becomes increasingly like z

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t table

• Each row is for a particular df
• Columns contain cumulative probabilities or tail
  regions
• Table contains t percentiles (like z scores)
• Notation tdf,p Example t9,.975 2.26

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95 CI for µ, s not known
Formula for a (1-a)100 confidence interval for µ
when s is NOT known
Same as z formula except replace z1-a/2 with
t1-a/2 and SEM with sem
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Illustrative example diabetic weight

• To what extent are diabetics over weight?
• Measure of ideal body weight (actual body
  weight) (ideal body weight) 100
• Data (n 18) 107, 119, 99, 114, 120, 104, 88,
  114, 124, 116, 101, 121, 152, 100, 125, 114, 95,
  117

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Interpretation of 95 CI for µ

• Remember that the CI seeks to capture µ, NOT
  x-bar
• 95 confidence means that 95 of similar
  intervals would capture µ (and 5 would not)
• For the diabetic body weight illustration, we can
  be 95 confident that the population mean is
  between 105.6 and 120.0

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Sample size requirements

• Assume SRS, Normality, valid data
• Let d ? the margin of error (half confidence
  interval length)
• To get a CI with margin of error d, use

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Sample size requirements, illustration

• Suppose, we have a variable with s 15

Smaller margins of error require larger sample
sizes
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Acronyms

• SRS ? simple random sample
• SDM ? sampling distribution of the mean
• SEM ? sampling error of mean
• CI ? confidence interval
• LCL ? lower confidence limit
• UCL ? lower confidence limit