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5:%20Introduction%20to%20estimation

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Estimate a statistic that 'guesstimates' a parameter ... Let a the probability confidence interval will not capture parameter. 1 a the confidence level ... – PowerPoint PPT presentation

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Title: 5:%20Introduction%20to%20estimation


1
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

2
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

3
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
4
Parameters and statistics
Parameters Statistics
Source Population Sample
Notation Greek (µ, s) Roman (x, s)
Random variable? No Yes
Calculated No Yes
5
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)

6
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

7
Results of simulation experiment
  • Findings
  • SDMs are centered on 29 (µ)
  • SDMs become tighter as n increases
  • SDMs become Normal as the n increases

8
95 Confidence Interval for µ
Formula for a 95 confidence interval for µ when
s is known
9
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
10
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
11
Interpretation of a 95 CI
We are 95 confident the parameter will be
captured by the interval.
12
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
13
(1 a)100 confidence for µ
Formula for a (1-a)100 confidence interval for µ
when s is known
14
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)

15
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
16
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)

17
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

18
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

19
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

20
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

21
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
22
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

23
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

24
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

25
Sample size requirements, illustration
  • Suppose, we have a variable with s 15

Smaller margins of error require larger sample
sizes
26
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
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