Economics 105: Statistics

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# Economics 105: Statistics - PowerPoint PPT Presentation

Economics 105: Statistics. Review # 1 due next Tuesday in class. Go over GH 7 & 8 No GH’s due until next Thur ! GH 9 and 10 due next Thur.

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### Economics 105: Statistics

Review #1 due next Tuesday in class.

Go over GH 7 & 8

No GH’s due until next Thur! GH 9 and 10 due next Thur.

Do go to lab still this week. It is due 2 weeks after your lab (so you’ll have 2 labs due that week, assuming you don’t complete it ahead of time)

### Sampling

Start with the Population, which is the set of all possible persons, firms, countries, etc. for the particular frame of reference

For each research question, define the relevant population:

what is the average income in the United States?

what is the average height in Krakozhia?

who will win the presidential election in November?

what is the average number of volunteer hours per student?

what percent of left-handed people have blue eyes?

A sample is the subset of the population selected for analysis

Must be representative of the population to avoid biased estimates

U.S. census taken every 10 years, according to the Constitution

First one in 1790 (3.9 million residents; today 312 million)

http://www.archives.gov/exhibits/charters/constitution_transcript.htmlhttp://www.ipums.org/? http://usa.ipums.org/usa-action/variables/group

### Simple Random Sampling

Most straightforward way to achieve representativeness is Simple Random Sampling where each person has an equal, and independent, chance of being selected

Also called i.i.d. sampling for independent and identically distributed (since drawn from same population)

Say we want to know how many magazines a household currently purchases. choose 1000 names from ________?

Suppose it is a good idea, now we contact them … if they’re not available, we just scratch them from our list. Or we go to the next name on the list until we find someone who is available. Any problems?

### Systematic Sample

Partition the population into n groups with k members each (k = N/n)

Randomly choose one from the first group of k

Take every kth item after that

Faster and easier than simple random sample

Telephone book, class roster, items from an assembly line, etc.

Greater chance of selection bias if there’s a pattern in the population

### Stratified Random Sampling

Hypothetical research question: What % of students will vote in the election?

Only have time & money to survey 100 students.

You do so, but get only 2 political science majors in your sample.

Problems?

Solution: Stratified Random Sampling

If a subgroup, or strata, of the population is particularly relevant to the research question, one may break the population down into strata and take a simple random sample from each strata

Each person can only belong to one strata

Ensures reasonable sample size of the subpopulation of interest or concern

Can stratify on > 1 characteristic -- major and gender

### Cluster Sampling

Hypothetical survey of rural families spread over a wide area

Hypothetical survey of homeless individuals in a large city

Problems?

Accurate list of population members

In-person interviews too costly

Mail surveys might lead to really high non-response

Solution: Cluster Sampling

Divide the population into geographically small units, or clusters

For example, political wards or residential blocks for a city

Then take a simple random sample of clusters

Each person or household in a chosen cluster is then contacted, that is, a complete census of chosen clusters

### Sources of Error from a Survey

Sampling Errors

come from having info on only a subset of population

statistical theory is used to quantify

Non-sampling Errors

can occur even with a complete census of the population

possible sources:

Population sampled is not relevant one or list is incomplete (coverage error, sample selection bias)

Measurement error

Halo effect

Poor wording of questions

Non-response (to whole survey or some questions)

try to minimize at outset & check up on some answers

### Sample Statistics

Population parameterSample statistic

### Sample Statistics

Denote an i.i.d. sample by X1, X2, X3, . . . ,Xn

What exactly is an Xi ?

Actual outcomes are x1, x2, x3, . . . , xn

How many samples could we take?

How many samples do we actually take?

A sample statisticis formed by taking some function of the random variables X1, X2, X3, . . . ,Xn, A = f(X1, X2, X3, . . . ,Xn)

Examples

The point estimate of the population parameter is a single number rather than a range

### Sampling Distribution

Sample statistic A is a random variable! Why?

Thus, a sample statistic has a probability distribution, known as a sampling distribution

Example: Let S = {0,1,2,3,4,5,6}

Graph the sampling distribution of for n = 2

### Central Limit Theorem

Rough statement of CLT:

“Sample means are eventually, approximately normally distributed.”

Formal statement of CLT:

Let X1, X2, X3, . . . ,Xn, where Xiis a random variable denoting the outcome of the ith observation, be an i.i.d. sample from ANY population distribution with mean and variance

then as n becomes large

Graphically (page 236 in BLK, 10th edition, has a nice visual)