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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 ofThe 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

HEIGHT

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Y.5

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Your Deviation Score Is Your Error Score

DEVIATION SCORE

100-10

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Y.5

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The More You Deviate From μ,The Less Likely Your Error

Deviation ErrorStandard Deviation Standard ErrorAverage Deviation Average ErrorExpected Deviation 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 less4.1” * 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 oftenExpect 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 MeansWill 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 PopulationPer 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 theStandard 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 MeanExpected Error of the MeanExpected 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 IsIs The SD Of The Sampling Distribution

•68% of the Sample Means will be within 1 SEM/SD of theGrand 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 thePopulation/Sample

As sample size Increases, Errors of Prediction DecreasePer 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 SEM SEMx2 (68%) (95%)

4.1” 1 4.1” 8.2” 4.1 2 (1.4) 2.9” 5.8” 4.1 3 (1.7) 2.4” 4.8” 4.1 4 (2) 2.05” 4.1” 4.1 9 (3) 1.37 2.74” 4.1 16 (4) 1.025 2.5” Or Less!

Sample Means have Probabilities Just Like Raw Scores

•The number of Standard Deviations X-bar is from μ

•The probability of a Sample with a Mean with (X-bar – μ) amountOf Error of Estimation

The Error of yourSample

The ExpectedError of yourSample

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

4. Expectation of Error Of Estimation

5. Evaluation

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 fromThe Specified PopulationBecause the Sample is too damn different from the Population

I, Anthropologist

I’m a Watusi, what’s it to ya??

I, Anthropologist

I’m a Watusi, what’s it to ya??

Watusi μ = 84 inchesWatusi = 4 inches

Watusi X-bar = 81.06 inches (H0)N = 6

Z = (81.1 – 84) / [4/(6)] = (-2.94)/1.63Z = -1.8

Critical Z (one-tail) for p < 0.05 = 1.65-1.8 > -1.65 Sample probably not Watusi

Reject Null Hypothesis!!!

I, Anthropologist

I’m a good player, Pygmy!!

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) / 1.84Z = 1.7

Critical Z (one-tail) for p < 0.05 = 1.651.7 > 1.65 Sample probably not Pygmy

Reject Null Hypothesis!!!

I, Anthropologist

Ubangi, u bet!!Have Mursi on me!!!!

I, Anthropologist

Ubangi, u bet!!Have Mursi on me!!!!

Ubangi μ = 70 inches Ubangi = 4.2 inches

Ubangi X-bar = 73.6 inches (H0)N = 6

Z = (73.6 – 70) / [4.2/(6)] = 3.6/1.71Z = 2.1

Critical Z (two-tail) for p < 0.05 = 1.962.1 > 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 SEMusing the Estimated Standard Deviation

My Ugly Cousin

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

Estimates

An Estimated SEM Has A Corresponding Estimated

Sampling DistributionFor an estimated value of the SEM:

The shape of the Sampling Distribution changes as N increasesThe 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 correspondingTo your sample size(A Z-table does not use N or Degrees of Freedom)

The t-TableOne-Tail

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: Estimate σ from the sample using N-1Step 2: Estimate SEM from Estimated σStep 3: Look up the 95% Confidence t-value two-tail) for N-1 dfStep 4: Multiple t-value (Step 2) by estimated SEM (Step 2)Step 5: Add Value from Step 3 to X-bar (UL: Upper Limit)Step 6: 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=25X-Bar = 70”Estimated σ = 3”Estimated SEM = 3/25 = 3/5 = 0.6”Critical t-value for 95% Confidence Interval (df=25-1) = 2.064Estimated 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”