Outline and Aims Review of Probability Theory Estimation Hypothesis Testing Confidence Intervals Summary
THE PROBABILITY FRAMEWORK FOR STATISTICAL
INFERENCE
1. The probability framework for statistical inference
a.留學生dissertation網Population, random variable, and distribution (see ProblemSet 1)
b. Moments of a distribution (mean, variance, standarddeviation, covariance, correlation; see PS1)
c. Conditional distributions and conditional means
d. Distribution of a sample of data drawn randomly from a
population: Y1; :::;Yn
2. Estimation
3. Testing
4. Confidence Intervals
Outline and Aims Review of Probability Theory Estimation Hypothesis Testing Confidence Intervals Summary
1C. CONDITIONAL DISTRIBUTIONS AND
CONDITIONAL MEANS
Conditional distributions
 The distribution of Y, given value(s) of some other random
variable, X.
 Example: the distribution of test scores, given that
STR < 20.
Conditional expectations and conditional moments
 conditional mean = mean of conditional distribution
= E(YjX = x) — important concept and notation
 conditional variance = variance of conditional distribution
 Example: E(Test scoresjSTR < 20) = the mean of test
scores among districts with small class sizes.
The difference in means is the difference between the means oftwo conditional distributions:Outline and Aims Review of Probability Theory Estimation Hypothesis Testing Confidence Intervals Summary
1D. DISTRIBUTION OF A SAMPLE OF DATA DRAWN
RANDOMLY FROM A POPULATION: Y1; :::;Yn
We will assume simple random sampling
 Choose and individual (district, entity) at random from the
population
Randomness and data
http://www.ttc-switch.com/Thesis_Writing/Economics/ Prior to sample selection, the value of Y is randombecause the individual selected is random
 Once the individual is selected and the value of Y isobserved, then Y is just a number. Is observed Y random?
NO.
 The data set is (Y1;Y2; :::;Yn), where Yi = value of Y for
http://www.ttc-switch.com/Thesis_Writing/Economics/the i-th individual (district, entity) sampledOutline and Aims Review of Probability Theory Estimation Hypothesis Testing Confidence Intervals Summary
ESTIMATION
Y is the natural estimator of the mean. But:
a. What are the properties of Y?
b. Why should we use rather than some other estimator?
 Y1 (the first observation)
 maybe unequal weights - not simple average
 median(Y1; :::;Yn)
留學生essayThe starting point is the sampling distribution of Y .Outline and Aims Review of Probability Theory Estimation Hypothesis Testing Confidence Intervals Summary#p#分頁標題#e#
(A) THE SAMPLING DISTRIBUTION OF Y
Y is a random variable, and its properties are determined by
the sampling distribution of Y
 The individuals in the sample are drawn at random.
 Thus the values of (Y1; :::;Yn) are random.
 Thus functions of (Y1; :::;Yn), such as Y, are random: had
a different sample been drawn, they would have taken on a
different value.
 The distribution of Y, over different possible samples of
size n, is called the sampling distribution of Y.
 The mean and variance of Y are the mean and variance of
its sampling distribution, E(Y) and var (Y).
 denoted by: Y  (E(Y); var (Y))
 The concept of the sampling distribution underpins all of
econometrics.Outline and Aims Review of Probability Theory Estimation Hypothesis Testing Confidence Intervals Summary
(b) Why Use Y To Estimate ?
1. Y is unbiased: E(Y) = 
2. Y is consistent: Y p