Download Estimation, Error, and Expectation

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Event, Estimation, Error, and
Expectation
Assume each one of you is a Sample of N=1:
•Your height (Xi) is the mean height of your sample
•Your height (mean height) is an Estimation of the Class (Population)
Mean height (μ, 67.6 inches)
•How much Error will your height (mean) have as an Estimator of
The class Mean height (μ)?
The Standard Deviation (4.1 inches) is a measure of the
Expected Error of Estimation
68% of the time you height will be off by no more that 4.1”
As an Estimator of μ (67.6 inches)
The More Extreme Your Are, The
Less Probable You Are
.5
PROBABILITY
.4
.3
.2
.1
0.0
55
60
70
65
HEIGHT
75
80
Your Deviation Score Is Your
Error Score
.5
PROBABILITY
.4
.3
.2
.1
0.0
-10
0
DEVIATION SCORE
10
The More You Deviate From μ,
The Less Likely Your Error
Deviation
Standard Deviation
Average Deviation
Expected Deviation
Error
Standard Error
Average Error
Expected Error
The probability of your height is the:
•Probability of your Deviation
•Probability of your Error of Estimation
95% of the time your Error of Estimation will be 2*SD or less
4.1” * 2 = 8.2
Think Many Samples
The farther off (Error) a Sample is (from μ):
•The Less frequent the Sample
•The Less frequent the Error
•The Less Probable the Error
Expect the Sample to occur less often
Expect the Error to be less Likely (Probability)
Sample Error of Estimation 
With Size
Central Limit Theorem:
•The sample means of an infinite number
Of samples (of the same size) from the same population will
Have a Grand Mean equal to μ
•The sample means will be normally distributed about μ
•The Standard Deviation of the Distribution of Sample Means
Will Decrease as Sample Size Increases
This means that the Errors of Estimation (X-bar- μ)
Will Decrease as the size of the samples Increases
Three Distributions
•Population Distribution:
Distribution of all raw scores in the population
•Sample Distribution:
Distribution of raw scores in a sample
•Sampling Distribution:
Distribution of infinite # of Sample Means (equal sample size)
from same Population
Per Grand Mean and CLT: Distribution of Sample Means is
Distribution of Errors of Estimation of μ
Standard Error Of The (Sample)
Mean
The Standard Deviation of the Sampling Distribution is the
Standard Error of the Mean
(I guarantee you that if you don’t know this I WILL fail you!!)
Standard Error of the Mean:
Average Error of the Mean
Expected Error of the Mean
Expected Error of the Sample Mean in Estimating μ
(See parentheses above!)
Sample Size and SEM
If all samples have N=1:
•The Sampling Distribution is Identical to the Sample
Distribution
•The Standard Error of the Mean equals the Standard Deviation
of the Sample
As the N of each sample increases for each (uniformly):
•Errors of Estimation Decrease
•Standard Error of the Mean Decreases
The SEM Is The SD Of The
Sampling Distribution
•68% of the Sample Means will be within 1 SEM/SD of the
Grand Mean/μ
•68% of the Sample Means will have Errors of Estimation (of μ)
of 1 SEM/SD or less.
•95% of the Sample Means will have Errors of Estimation (of μ)
of 2 SEM/SDs or less
Computing The SEM/SD Of The
Sampling Distribution
With sample sizes N=1, the SEM equals the SD of the
Population/Sample
As sample size Increases, Errors of Prediction Decrease
Per N in the denominator
The SEM/SD Is Determined By
Two Factors
1. Heterogeneity of the
Parent Population
2. Size of the Samples
Computing The SEM/SD Of The
Sampling Distribution
SD
N
4.1”
4.1
4.1
4.1
4.1
4.1
1
2 (1.4)
3 (1.7)
4 (2)
9 (3)
16 (4)
SEM SEMx2
(68%) (95%)
4.1”
8.2”
2.9”
5.8”
2.4”
4.8”
2.05” 4.1”
1.37
2.74”
1.025 2.5”
Or Less!
Sample Means have Probabilities
Just Like Raw Scores
The Error of your
Sample
The Expected
Error of your
Sample
•The number of Standard Deviations X-bar is from μ
•The probability of a Sample with a Mean with (X-bar – μ) amount
Of Error of Estimation
Error of Estimation
Because of the Shape of the Bell Curve of Sampling Distributions:
•Probability of Error decreases with Size of Error
•Versus a rectangular distribution
The 5 Es of Experimentation
2. Estimation
3. Error of Estimation
5. Evaluation
4. Expectation of Error
Of Estimation
Every Sample Has a Z
A Sample’s Z tells you the probability of:
•Getting this Sample from the Specified Population
•Finding a Sample with a Mean this far away from the Specified μ
•Making a mistake if you decide that this Sample didn’t come from
The Specified Population
Because the Sample is too damn different from the Population
I, Anthropologist
I’m a Watusi, what’s it to ya??
I, Anthropologist
Watusi μ = 84 inches
Watusi  = 4 inches
 Watusi X-bar = 81.06 inches (H0)
N=6
Z = (81.1 – 84) / [4/(6)] = (-2.94)/1.63
Z = -1.8
Critical Z (one-tail) for p < 0.05 = 1.65
-1.8 > -1.65  Sample probably not Watusi
Reject Null Hypothesis!!!
I’m a Watusi, what’s it to ya??
I, Anthropologist
 Pygmy μ = 42 inches
 Pygmy  = 4.5 inches
 Pygmy X-bar = 45.12 inches (H0)
N=6
Z = (45.12 – 42) / [4.5/(6)] = (3.12) / 1.84
Z = 1.7
Critical Z (one-tail) for p < 0.05 = 1.65
1.7 > 1.65  Sample probably not Pygmy
Reject Null Hypothesis!!!
I’m a good player, Pygmy!!
I, Anthropologist
Ubangi, u bet!!
Have Mursi on me!!!!
I, Anthropologist
 Ubangi μ = 70 inches
 Ubangi  = 4.2 inches
 Ubangi X-bar = 73.6 inches (H0)
N=6
Z = (73.6 – 70) / [4.2/(6)] = 3.6/1.71
Z = 2.1
Critical Z (two-tail) for p < 0.05 = 1.96
Ubangi, u bet!!
Have Mursi on me!!!! 2.1 > 1.96  Sample probably not Ubangi
Reject Null Hypothesis!!!
z, t, F =
z, t, F =
What Do You Do If You Don’t
Know σ?
Beer And Western Civilization
•Louis Pasteur
pasteurization
•Carlsberg Brew Master
pH system
•Guinness Statistician (Gosset)
t-test
What Do You Do If You Don’t
Know σ?
If you want to evaluate the probability of a Sample (Mean):
If you know μ but you don’t know σ:
1. Estimate σ from the sample using N-1
2. Then Estimate the SEM
using the Estimated Standard Deviation
My Ugly Cousin
Estimates
An Estimated SEM will have some Error:
The larger the sample size:
•The less error in estimating the population SD
•Hence, the less error in estimating the SEM
An Estimated SEM Has A
Corresponding Estimated
Sampling Distribution
For an estimated value of the SEM:
The shape of the Sampling Distribution changes as N increases
The SD/SEM is Wider than SN-Curve the Smaller the Sample N
There Is A Separate Estimated
Sampling Distribution For Every
Sample N
Look up the probability of t in the t-table
The t-Table Is A Table Of Tables
A separate table for every sample size:
•Degrees of Freedom: df = N-1
•Use Row in t-table with Degrees of Freedom corresponding
To your sample size
(A Z-table does not use N or Degrees of Freedom)
One-Tail
The t-Table
Two-Tail
Going From Sample To
Population
If you have a Sample Mean, this is the best estimate of μ
How Confident can you be about your Estimate?
Step 1:
Step 2:
Step 3:
Step 4:
Step 5:
Step 6:
Estimate σ from the sample using N-1
Estimate SEM from Estimated σ
Look up the 95% Confidence t-value two-tail) for N-1 df
Multiple t-value (Step 2) by estimated SEM (Step 2)
Add Value from Step 3 to X-bar (UL: Upper Limit)
Subtract Value from Step 4 from X-bar (LL: Lower Limit)
μ has a 95% chance of being between UL & LL
Free One-Size Fits All Pants For
The Men of UMD
How tall is the Average (μ) UMD Male?
N=25
X-Bar = 70”
Estimated σ = 3”
Estimated SEM = 3/25 = 3/5 = 0.6”
Critical t-value for 95% Confidence Interval (df=25-1) = 2.064
Estimated SEM * Critical t-value = 0.6” * 2.064 = 1.2384”
UL = X-bar + (Estimated SEM * Critical t) = 70” + 1.24” = 71.24”
LL = X-bar - (Estimated SEM * Critical t) = 70” – 1.24” = 68.76”