This document provides an introduction to statistical estimation. It discusses sampling distributions, parameters and statistics, confidence intervals when the population standard deviation is both known and unknown, and sample size requirements. When the population standard deviation is unknown, Student's t-distribution must be used instead of the normal distribution to account for additional variability. Examples are provided to illustrate key concepts like confidence intervals, margins of error, and how sample size relates to desired precision of estimates.

1. 11/9/2022 5: Intro to estimation 1
5: Introduction to estimation
(A) Intro to statistical inference
(B) Sampling distribution of the mean
(C) Confidence intervals (σ known)
(D) Student’s t distributions
(E) Confidence intervals (σ not known)
(F) Sample size requirements

2. 11/9/2022 5: Intro to estimation 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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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, σ) 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, it’s
“behavior” is predictable
The behavior of x-bar is predicted by its
sampling distribution, the Sampling
Distribution of the Mean (SDM)

6. 11/9/2022 5: Intro to estimation 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
AGE
65.0
60.0
55.0
50.0
45.0
40.0
35.0
30.0
25.0
20.0
15.0
10.0
5.0
0.0
200
100
0

7. 11/9/2022 5: Intro to estimation 7
Results of simulation experiment
Findings:
(1) SDMs are
centered on
29 (µ)
(2) SDMs
become
tighter as n
increases
(3) SDMs
become
Normal as
the n
increases

8. 11/9/2022 5: Intro to estimation 8
95% Confidence Interval for µ
n
SEM
SEM
x
s


where
)
)(
96
.
1
(
Formula for a 95% confidence interval for μ when σ is known:

9. 11/9/2022 5: Intro to estimation 9
Example
 Population with σ = 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)
Illustrative example
Margin of error

10. 11/9/2022 5: Intro to estimation 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
margin of error
point estimate

11. 11/9/2022 5: Intro to estimation 11
Interpretation of a 95% CI
We are 95% confident the parameter will be captured by the interval.

12. 11/9/2022 5: Intro to estimation 12
Other levels of confidence
Confidence level
1 – a
Alpha level
a
z1–a/2
.90 .10 1.645
.95 .05 1.96
.99 .01 2.58
Let a  the probability confidence interval will not capture parameter
1 – a  the confidence level

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(1 – a)100% confidence for μ
SEM
z
x 
  2
1 a
Formula for a (1-α)100% confidence interval for μ when σ is known:

14. 11/9/2022 5: Intro to estimation 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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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
 σ known ahead of time
It does not account for:
 GIGO
 Poor quality samples (e.g., due to non-
response)

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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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Student’s t distributions
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 σ
Brilliant!

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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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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, σ not known
n
s
sem
sem
t
x n


 

where
2
1
,
1 a
Same as z formula except replace z1a/2 with t1a/2 and SEM with sem
Formula for a (1-α)100% confidence interval for μ when σ is NOT known:

22. 11/9/2022 5: Intro to estimation 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}
120.0)
(105.6,
=
7.17
±
112.778
)
44
.
3
)(
110
.
2
(
778
.
112
)
)(
(
table)
(from
110
.
2
400
.
3
18
242
.
14
424
.
14
778
.
112
2
2
05
.
2
1
,
1
975
,.
17
1
,
1
18
1
,
1


















sem
t
x
t
t
t
t
n
s
sem
s
x
n
n
a
a

23. 11/9/2022 5: Intro to estimation 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. 11/9/2022 5: Intro to estimation 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:
2
2
4
d
n
s


25. 11/9/2022 5: Intro to estimation 25
Sample size requirements, illustration
Suppose, we have a variable with s = 15
36
5
15
4
use
,
5
For 2
2



 n
d
144
5
.
2
15
4
use
,
5
.
2
For 2
2



 n
d
900
1
15
4
use
,
1
For 2
2



 n
d
Smaller margins of
error require larger
sample sizes

26. 11/9/2022 5: Intro to estimation 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