1. Statistical Inference

2. Concepts (Frequentist / Classical)
Population:
μ
σ
μX − μY
σX/σY
D
Sample:
X
s
X − Y
sX/sY
Δ
Constant parameters
(unknown)
Random variables (known but
depends on sample being drawn)
… used to infer …

3. Concepts
Population:
μ
Sample:
X
Want to test if μ is equal to μ0 (μ0 is constant)
H0: μ = μ0 H1: μ ≠ μ0

4. Concepts
H0: μ = μ0
Rejected if X is “very far” from μ0
μ0
More likely to reject H0More likely to reject H0
… as X got closer to the
corner
As X got closer to the
corner
… as the area to the
corner decreases
… as the area to the
corner decreases
X − μ0
σ/ n
−
X − μ0
σ/ n
How “far” is “very far”?
P–value

5. If α is very large
… depends on the threshold, α
μ0
X − μ0
σ/ n
−
X − μ0
σ/ n
Even something this
close to μ0 is
considered “far
enough” to reject H0
Blue = α
Red = P–value

6. If α is very small
… depends on the threshold, α
μ0
X − μ0
σ/ n
−
X − μ0
σ/ n
Must be this far from μ0
to be considered “far
enough” to reject H0
Blue = α
Red = P–value

7. P–value
Concepts
α Set by the experimenter
Determined by the data / sample

8. Reject H0 only if P − value ≤ α
X − μ0
σ/ n
−
X − μ0
σ/ 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 = α
Red = P–value

9. H0 is considered plausible / is not
rejected if P − value > α
X − μ0
σ/ n
−
X − μ0
σ/ 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 = α
Red = P–value

10. Want to test if μ is > μ0 (μ0 is constant)
H0: μ ≤ μ0 H1: μ > μ0
Concepts
Rejected if X is “much larger” than μ0
μ0
Blue = α
Red = P–value

11. Want to test if μ is < μ0 (μ0 is constant)
H0: μ ≥ μ0 H1: μ < μ0
Concepts
Rejected if X is “much smaller” than μ0
μ0
Blue = α
Red = P–value

12. Type I Error, Type II Error and power
μ0
H0: μ ≤ μ0
H1: μ > μ0
(specifically, μ = μ1)
μ1
max P Type I error = max P reject|H0 is true = α
min P Type II error = min P fail to reject|H0 is false = min P fail to reject|H1 is true = β
max power = max P reject|H0 is false = max P reject|H1 is true = 1 − β

13. Statistical Testing in a Nutshell
This is what is plotted on the distribution curve

14. Statistical Testing in a Nutshell

15. Testing population standard deviation
Want to test if σ is less than σ0 (σ0 is constant)
H0: σ = σ0 H1: σ < σ0
Rejected if s is “much smaller” than σ0 … or if χ2
= n − 1
s2
σ0
2 < χ1−α
2

16. … or if χ2
= n − 1
s2
σ0
2 < χ1−α
2
From the data / experiment
From table
Suppose n = 6 and 1 − α = 0.05
Then ν = 6 − 1 = 5

17. Want to test if σ is greater than σ0 (σ0 is constant)
H0: σ = σ0 H1: σ > σ0
Rejected if s is “much larger” than σ0 … or if χ2
= n − 1
s2
σ0
2 > χα
2
Suppose n = 6 and α = 0.05
Then ν = 6 − 1 = 5

18. 1–way ANOVA

19. What is likely to come up in a closed–
laptop exam?
Completing an ANOVA table
Interpreting an ANOVA table

20. 1–way ANOVA
a levels or treatments
n replicates at EACH level / treatment
Goal: H0: μ1 = μ2 = ⋯ = μa

21. 1–way ANOVA
Always relative to MSE
Always SS divided by Degrees
of Freedom (DOF)
Explained variation
Total variation

22. 2–way ANOVA
3 levels or treatments for row factor (a)
4 replicates at EACH treatment combination (n)3 levels or treatments for column factor (b)

23. 2–way ANOVA
Always relative to MSEExplained variation
Total variation

24. Reference
Navidi, William Cyrus. Statistics for engineers and
scientists. Vol. 1. New York: McGraw-Hill, 2006