Statistics Presentation (sample)
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Transcript of Statistics Presentation (sample)
Statistical Inference
Concepts (Frequentist / Classical)
Population: Sample:
Constant parameters (unknown)
Random variables (known but depends on sample being drawn)
… used to infer …
Concepts
Population: Sample:
Want to test if is equal to ( is constant)
H 0 : μ=μ0 H 1 : μ≠μ0
ConceptsH 0 : μ=μ0
Rejected if is “very far” from μ0
μ0
More likely to reject H0More likely to reject H0
… as got closer to the corner
As got closer to the corner
… as the area to the corner decreases
… as the area to the corner decreases
X −μ0σ / √n
−X −μ0σ /√n
How “far” is “very far”?
P–value
If α is very large
… depends on the threshold, α
μ0 X −μ0σ / √n
−X −μ0σ /√n
Even something this close to μ0 is considered “far enough” to reject H0
Blue = αRed = P–value
If α is very small
… depends on the threshold, α
μ0 X −μ0σ / √n
−X −μ0σ /√n
Must be this far from μ0 to be considered “far enough” to reject H0
Blue = αRed = P–value
P–value
Concepts
α Set by the experimenter
Determined by the data / sample
Reject H0 only if
X −μ0σ / √n
−X −μ0σ /√n
μ0
You only have to go this far to reject H0
… but your data is even further away than that (i.e. more extreme) So, reject H0
Blue = αRed = P–value
H0 is considered plausible / is not rejected if
X −μ0σ / √n−
X −μ0σ /√n
μ0
You have to go this far to reject H0
… but your data is not as far as that (i.e. less extreme)
So, fail to reject H0
Blue = αRed = P–value
Want to test if is > ( is constant)
H 0 : μ≤ μ0 H 1 : μ>μ0
Concepts
Rejected if is “much larger” than μ0 μ0
Blue = αRed = P–value
Want to test if is < ( is constant)
H 0 : μ≥ μ0 H 1 : μ<μ0
Concepts
Rejected if is “much smaller” than μ0 μ0
Blue = αRed = P–value
Type I Error, Type II Error and power
μ0H 0 : μ≤ μ0 (specifically, ) μ1
max P (Type I error )=max P (reject∨H 0 istrue )=αmin P (Type II error )=min P ( fail¿ reject∨H 0is false )=minP ( fail¿reject∨H 1 istrue )=βmax power=max P (reject∨H 0 is false )=max P (reject∨H 1is true )=1− β
Statistical Testing in a NutshellThis is what is plotted on the distribution curve
Statistical Testing in a Nutshell
Testing population standard deviation
Want to test if is less than ( is constant)
H 0 : σ=σ0 H 1 :σ<σ 0
Rejected if is “much smaller” than σ0… or if
… or if
From the data / experiment
From table
Suppose and
Then
Want to test if is greater than ( is constant)
H 0 : σ=σ0 H 1 :σ>σ 0
Rejected if is “much larger” than σ0… or if
Suppose and
Then
1–way ANOVA
What is likely to come up in a closed–laptop exam?
Completing an ANOVA tableInterpreting an ANOVA table
1–way ANOVA
levels or treatments
replicates at EACH level / treatment
Goal:
1–way ANOVAAlways relative to MSE
Always SS divided by Degrees of Freedom (DOF)
Explained variation
Total variation
2–way ANOVA
levels or treatments for row factor (a)
replicates at EACH treatment combination (n) levels or treatments for column factor (b)
2–way ANOVA
Always relative to MSEExplained variation
Total variation
Reference
Navidi, William Cyrus. Statistics for engineers and scientists. Vol. 1. New York: McGraw-Hill, 2006