1. POPULATION (of “units”)
uniform
X = “Random Variable”
symmetric unimodal
“REAL WORLD” SYSTEM
skew (positive)
PROBABILTY MODEL
THEORY
EXPERIMENT
Is there a significant difference?
Random Sample
Model Predictions
STATISTICS
How do we test them?

2. POPULATION (of “units”)
uniform
X = “Random Variable”
symmetric unimodal
“REAL WORLD” SYSTEM
To be a valid pdf, f(x) must satisfy two conditions:
f (x)  0
Total area under its graph = 1.
skew (positive)
PROBABILTY DENSITY CURVE
PROBABILTY DENSITY FUNCTION (pdf)

3. POPULATION (of “units”)
uniform
X = “Random Variable”
symmetric unimodal
“REAL WORLD” SYSTEM
To be a valid pdf, f(x) must satisfy two conditions:
f (x)  0
Total area under its graph = 1.
skew (positive)
PROBABILTY DENSITY CURVE
PROBABILTY DENSITY FUNCTION (pdf)

4. POPULATION (of “units”)
uniform
X = “Random Variable”
symmetric unimodal
“REAL WORLD” SYSTEM
To be a valid pdf, f(x) must satisfy two conditions:
f (x)  0
Total area under its graph = 1.
skew (positive)
PROBABILTY DENSITY CURVE
PROBABILTY DENSITY FUNCTION (pdf)

5. POPULATION (of “units”)
uniform
X = “Random Variable”
symmetric unimodal
“REAL WORLD” SYSTEM
To be a valid pdf, f(x) must satisfy two conditions:
f (x)  0
Total area under its graph = 1.
skew (positive)
PROBABILTY DENSITY CURVE
PROBABILTY DENSITY FUNCTION (pdf)

6. POPULATION (of “units”)
uniform
X = “Random Variable”
symmetric unimodal
“REAL WORLD” SYSTEM
To be a valid pdf, f(x) must satisfy two conditions:
f (x)  0
Total area under its graph = 1.
skew (positive)
PROBABILTY DENSITY CURVE
PROBABILTY DENSITY FUNCTION (pdf)
CUMULATIVE DISTRIBUTION FUNCTION (cdf)
increases from 0 to 1

7. POPULATION (of “units”)
uniform
X = “Random Variable”
symmetric unimodal
“REAL WORLD” SYSTEM
To be a valid pdf, f(x) must satisfy two conditions:
f (x)  0
Total area under its graph = 1.
skew (positive)
PROBABILTY DENSITY CURVE
PROBABILTY DENSITY FUNCTION (pdf)
CUMULATIVE DISTRIBUTION FUNCTION (cdf)
INTERVAL PROBABILITY
Fundamental Thm of Calculus

8. ~ The Normal Distribution ~ (a.k.a. “The Bell Curve”)X
Johann Carl Friedrich Gauss
standard deviation
X ~ N(μ, σ) σ
mean μ

9. Standard Normal Distribution
Z ~ N(0, 1)
SPECIAL CASE
Total Area = 1 1 Z
The cumulative distribution function (cdf) is denoted by (z). It is not expressible in explicit, closed form, but is tabulated, and computable in R via the command pnorm.

10. Standard Normal Distribution
Example
Standard Normal Distribution
Z ~ N(0, 1)
Find (1.2) = P(Z  1.2).
Total Area = 1 1 Z
1.2
“z-score”

11. Standard Normal Distribution
Example
Standard Normal Distribution
Z ~ N(0, 1)
Find (1.2) = P(Z  1.2).
Use the included table.
Total Area = 1 1 Z
1.2
“z-score”

12. Lecture Notes Appendix…

14. Standard Normal Distribution
Example
Standard Normal Distribution
Z ~ N(0, 1)
Find (1.2) = P(Z  1.2).
Use the included table.
Use R:
> pnorm(1.2)
[1]
Total Area = 1 1
P(Z > 1.2) Z
1.2
“z-score”
Note: Because this is a continuous distribution, P(Z = 1.2) = 0, so there is no difference between P(Z > 1.2) and P(Z  1.2), etc.

15. Standard Normal Distribution
Z ~ N(0, 1) μ σ
X ~ N(μ, σ) 1 Z
Why be concerned about this, when most “bell curves” don’t have mean = 0, and standard deviation = 1?
Any normal distribution can be transformed to the standard normal distribution via a simple change of variable.

16. Random Variable
X = Age at first birth
POPULATION
Example
Question: What proportion of the population had their first child before the age of 27.2 years old?
P(X < 27.2) = ?
Year 2010
X ~ N(25.4, 1.5)
μ = 25.4
σ = 1.5
27.2

17. Random Variable POPULATION Example X ~ N(25.4, 1.5)
X = Age at first birth
POPULATION
Example
Question: What proportion of the population had their first child before the age of 27.2 years old?
P(X < 27.2) = ?
The x-score = 27.2 must first be transformed to a corresponding z-score.
Year 2010
X ~ N(25.4, 1.5)
σ = 1.5
μ = 25.4
μ = 25.4
μ =
27.2 33

18. Random Variable
X = Age at first birth
POPULATION
Example
Question: What proportion of the population had their first child before the age of 27.2 years old?
P(X < 27.2) = ?
P(Z < 1.2) =
Year 2010
X ~ N(25.4, 1.5)
σ = 1.5
Using R:
> pnorm(27.2, 25.4, 1.5)
[1]
μ = 25.4
μ =
27.2 33

19. Standard Normal Distribution
Z ~ N(0, 1) 1 Z
What symmetric interval about the mean 0 contains 95% of the population values?
That is…

20. Standard Normal Distribution
Z ~ N(0, 1)
Use the included table.
0.95
0.025
0.025 Z
-z.025 = ?
+z.025 = ?
What symmetric interval about the mean 0 contains 95% of the population values?
That is…

21. Lecture Notes Appendix…

23. Standard Normal Distribution
Z ~ N(0, 1)
Use the included table.
Use R:
> qnorm(.025)
[1]
> qnorm(.975)
[1]
0.95
0.025
0.025 Z
-z.025 = -1.96
-z.025 = ?
“.025 critical values”
+z.025 = +1.96
+z.025 = ?
What symmetric interval about the mean 0 contains 95% of the population values?

24. Standard Normal Distribution
Z ~ N(0, 1)
X ~ N(25.4, 1.5)
X ~ N(μ, σ)
What symmetric interval about the mean age of 25.4 contains 95% of the population values?
22.46  X  yrs
> areas = c(.025, .975)
> qnorm(areas, 25.4, 1.5)
[1]
0.95
0.025
0.025 Z
-z.025 = -1.96
-z.025 = ?
“.025 critical values”
+z.025 = +1.96
+z.025 = ?
What symmetric interval about the mean 0 contains 95% of the population values?

25. Standard Normal Distribution
Z ~ N(0, 1)
Use the included table.
0.90
0.05
0.05 Z
-z.05 = ?
+z.05 = ?
Similarly…
What symmetric interval about the mean 0 contains 90% of the population values?

26. …so average 1.64 and 1.65
0.95  average of and …

27. Standard Normal Distribution
Z ~ N(0, 1)
Use the included table.
Use R:
> qnorm(.05)
[1]
> qnorm(.95)
[1]
0.90
0.05
0.05 Z
-z.05 = ?
-z.05 =
+z.05 =
+z.05 = ?
“.05 critical values”
Similarly…
What symmetric interval about the mean 0 contains 90% of the population values?

28. In general…. Normal Distribution
What symmetric interval about the mean contains 100(1 – )% of the population values?
1 – 
 / 2
 / 2
“ / 2 critical values”
Example:

29. Use the included table or R:
In general….
Normal Distribution
What symmetric interval about the mean contains 100(1 – )% of the population values?
“Approximately 95% of any normally-distributed population lies within 2 standard deviations of the mean.”
“.025 critical values”
“ / 2 critical values”
cumulative areas
Example:
Use the included table or R:
> qnorm(c(.025, .975))
[1]

30. POPULATION (of “units”)
uniform
X = “Random Variable”
symmetric unimodal
“REAL WORLD” SYSTEM
skew (positive)
PROBABILTY DENSITY FUNCTION (pdf)
Population Distribution of X
Standard
Deviation

31. POPULATION (of “units”)
uniform
X = “Random Variable”
symmetric unimodal
“REAL WORLD” SYSTEM
skew (positive)
PROBABILTY DENSITY FUNCTION (pdf)
Population Distribution of X
Sample 1,
size n
Standard
Deviation
Sample 3,
size n
Sample 2,
size n
ad infinitum…..

32. POPULATION (of “units”)
uniform
X = “Random Variable”
symmetric unimodal
“REAL WORLD” SYSTEM 1
“Standard Error”
skew (positive)
PROBABILTY DENSITY FUNCTION (pdf)
Population Distribution of X
Standard
Deviation
CENTRAL LIMIT
THEOREM