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