1

Lecture 2

This lecture introduces you to

1. Review of probability / statistics

2. Topic: What is the effect on test scores of

reducing class size?

3. Introduction to STATA

Visit http://eco323fall08.wordpress.com/ for class

information and more!

2

Review of Probability and Statistics

(SW Chapters 2, 3)

• How much do you remember from Eco230?

o Q1: Let X be the random variable distributed as Normal

(5,4). Find the probabilities of the following events

� P(X ≤ 6)

� P(X > 4)

3

Moments of a population distribution: mean, variance, standard deviation

mean = expected value (expectation) of Y

= E(Y)

= µY

variance = E(Y – µY)2

= 2

Yσ

standard deviation = variance = σY

covariance between X and Z = cov(X,Z)

= E[(X – µX)(Z – µZ)] = σXZ

cf. corr(X,Z) = cov( , )

var( ) var( )

XZ

X Z

X Z

X Z

σσ σ

= = rXZ

4

Review of Probability and Statistics

(SW Chapters 2, 3)

• How much do you remember from Eco230?

o Q1: Let X be the random variable distributed as Normal

(5,4). Find the probabilities of the following events

� P(X ≤ 6)

� P(X ≥ 4)

5

The mean and variance of the

sampling distribution of Y

mean: E(Y ) = E(1

1 n

i

i

Yn =∑ ) =

1

1( )

n

i

i

E Yn =∑ =

1

1 n

Y

inµ

=∑ = µY

Variance: var(Y ) = E[Y – E(Y )]2

= E[Y – µY]2

= E

2

1

1 n

i Y

i

Yn

µ=

−

∑

= E

2

1

1( )

n

i Y

i

Yn

µ=

−

∑

6

so var(Y ) = E

2

1

1( )

n

i Y

i

Yn

µ=

−

∑

= 1 1

1 1( ) ( )

n n

i Y j Y

i j

E Y Yn n

µ µ= =

− × −

∑ ∑

= 2

1 1

1( )( )

n n

i Y j Y

i j

E Y Yn

µ µ= =

− − ∑∑

= 2

1 1

1cov( , )

n n

i j

i j

Y Yn = =∑∑

= 2

21

1 n

Y

inσ

=∑

= 2

Y

n

σ

7

Sampling distribution of when Y

is Bernoulli, p = 0.78:

Y

8

The Central Limit Theorem (CLT):

If (Y1,…,Yn) are i.i.d. and 0 < 2

Yσ < ∞, then when n is large

the distribution of Y is well approximated by a normal

distribution N(µY, 2

Y

n

σ).

→That is, ( )

var( )

Y E Y

Y

− =

/

Y

Y

Y

n

µσ−

is approximately distributed

as N(0,1)

9

Review of Probability and Statistics

o Q2: You are hired by the governor to study whether a tax

on liquor has decreased average liquor consumption in

your state.

o You are able to obtain the difference in liquor

consumption Y (in ounces) for the years before and after

the tax. Treat this as a random sample from a Normal (µ,

σ2) distribution

10

Review of Probability and Statistics

(SW Chapters 2, 3)

o Q2: You are hired by the governor to study whether a tax on liquor

has decreased average liquor consumption in your state. You are

able to obtain the difference in liquor consumption Y (in ounces) for

the years before and after the tax. Treat this as a random sample

from a Normal (µ, σ2) distribution

(a)The null hypothesis is that there was no change in

average liquor consumption. State this formally in terms

of µ

11

Review of Probability and Statistics

(SW Chapters 2, 3)

o Q2: You are hired by the governor to study whether a tax on liquor

has decreased average liquor consumption in your state. You are

able to obtain the difference in liquor consumption Y (in ounces) for

the years before and after the tax. Treat this as a random sample

from a Normal (µ, σ2) distribution

(b)The alternative is that there was a decline in liquor

consumption. State the alternatives in terms of µ

12

Review of Probability and Statistics

(SW Chapters 2, 3)

o Q2: You are hired by the governor to study whether a tax on liquor

has decreased average liquor consumption in your state. You are able

to obtain the difference in liquor consumption Y (in ounces) for the

years before and after the tax. Treat this as a random sample from a

Normal (µ, σ2) distribution

(c)Now suppose your sample size is n = 900 and you

obtain the estimates y_bar=-32.8 and s=466.4. Calculate

the test statistic for testing H0 against H1.