1. 5: Introduction to estimation
Intro to estimation (confidence intervals)
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5: Introduction to estimation
Intro to statistical inference
Sampling distribution of the mean
Confidence intervals (σ known)
Student’s t distributions
Confidence intervals (σ not known)
Sample size requirements
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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
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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
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4. Parameters and statistics
Source
Population
Sample
Notation
Greek (μ, σ)
Roman (x, s)
Random variable? No
Yes
Calculated
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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, it’s “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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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
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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
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8. 95% Confidence Interval for µ
Formula for a 95% confidence interval for μ when σ is known:
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9. Illustrative example Example SEM = s / n = 13.586 / 10 = 4.30
Population with σ = (known ahead of time)
SRS  {21, 42, 11, 30, 50, 28, 27, 24, 52}
n = 10, x-bar = 29.0
SEM = s / n = / 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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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
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11. Interpretation of a 95% CI
We are 95% confident the parameter will be captured by the interval.
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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
z1–a/2
.90
.10
1.645
.95
.05
1.96
.99
.01
2.58
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13. (1 – a)100% confidence for μ
Formula for a (1-α)100% confidence interval for μ when σ is known:
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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)
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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
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16. Beware Prior CI formula applies only to It does not account for: SRS
Normal SDMs
σ known ahead of time
It does not account for:
GIGO
Poor quality samples (e.g., due to non-response)
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17. When σ is Not Known In practice we rarely know σ
Instead, we calculate s and use this as an estimate of σ
This adds another element of uncertainty to the inference
A modification of z procedures called Student’s t distribution is needed to account for this additional uncertainty
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18. Student’s t distributions
Brilliant!
William Sealy Gosset ( ) 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 σ
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19. t Distributions
Student’s 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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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
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21. Same as z formula except replace z1-a/2 with t1-a/2 and SEM with sem
95% CI for µ, σ not known
Formula for a (1-α)100% confidence interval for μ when σ is NOT known:
Same as z formula except replace z1-a/2 with t1-a/2 and SEM with sem
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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}
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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 and 120.0
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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:
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25. Sample size requirements, illustration
Suppose, we have a variable with s = 15
Smaller margins of error require larger sample sizes
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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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