Topics Semester I Descriptive statistics Time series Semester II Sampling Statistical Inference: Estimation, Hypothesis testing Relationships, casual models

Sampling

Statistical observations Expectations: Quickness Accuracy Reliability Solutions Observe each individuals Sampling

Statistical inference Descriptive statistics: Describe the observed elements Statistical inference: Inferences to the populations which are based on the sample. (Estimation, hypothesis testing)

Estimation: Estimate the population parameter from a sample Types Point Interval

Inferential population Target population Coverage error Sampling frame Sampling error Sample Nonresponse error RespondentsConstruct Validity Measurement Measurement error Response Processing error Edited dataSurvey statistics Error types

Type of Errors Sampling error: due to selecting a sample instead of the entire population Nonsampling error: errors due to mistakes

Issues Probability vs. Nonprobability samples Sample size Representativity

Probability versus Nonprobability Probability Samples: each member of the population has a known non-zero probability of being selected Methods include random sampling, systematic sampling, and stratified sampling. Nonprobability Samples: members are selected from the population in some nonrandom manner Methods include convenience sampling, judgment sampling, quota sampling, and snowball sampling

Random Sampling Random sampling is the purest form of probability sampling. Simple Random sample with replacement: Each member of the population has an equal and known chance of being selected. Simple Random sample without replacement

Stratified Sampling Stratified sampling is commonly used probability method that is superior to random sampling because it reduces sampling error. A stratum is a subset of the population that share at least one common characteristic; such as males and females. Identify relevant stratums and their actual representation in the population. Random sampling is then used to select a sufficient number of subjects from each stratum. Stratified sampling is often used when one or more of the stratums in the population have a low incidence relative to the other stratums.

Cluster Sampling Cluster Sample: a probability sample in which each sampling unit is a collection of elements. Effective under the following conditions: A good sampling frame is not available or costly, while a frame listing clusters is easily obtained The cost of obtaining observations increases as the distance separating the elements increases Examples of clusters: City blocks – political or geographical Housing units – college students Hospitals – illnesses Automobile – set of four tires

Snowball Sampling Snowball sampling is a special nonprobability method used when the desired sample characteristic is rare. It may be extremely difficult or cost prohibitive to locate respondents in these situations. This technique relies on referrals from initial subjects to generate additional subjects. It lowers search costs; however, it introduces bias because the technique itself reduces the likelihood that the sample will represent a good cross section from the population.

We examine - Distribution of variables and parameters - Relationship between variables SamplePopulation SizenN Mean Std. Dev. s* ProprtionP

Estimation: Estimate the population parameter from a sample Types Point Interval

Point estimation

The statistic is computed from sample to estimate the population parameter Consistence

Estimation of population mean Can the sample mean be a potential estimation? Yes, if

Example Population:10, 11, 12, 13, 14 Mean (  ): 12 Variance (  2 ): 2 Std. Dev. (  ): 1, Size (N):5 Sample size (n):2 Consider each sample with sample size 2 Describe the distribution of the sample means! Calculate the expected value of the sample means! Sample distribution: distribution of the examined parameter.

What is the result? The expected value of the sample means with given sample size is equal to the population mean

Point estimation of pop. STD. DEV. Corrected empirical std. Dev.

Point estimation of proportion With replacement Without replacement

Standard Error of the estimation The difference on average between the sample statistics and the population parameter with given sample size

In the case of the sample means: he standard error of the estimation The difference on average between the sample means and the population mean. Standard error of the mean

Calculation What is happened if  is unknown? With replacement Without replacement

With replacement Without replacement Estimation

INTERVAL ESTIMATE OF THE POPULATION MEAN

Structure of the confidence interval 95%s interval: from 100 estimates on average 95 contain the population mean First step? Point estimation Maximum error: with a given probability the maximum error of the estimation

!!!! Maximum error: with a given probability the maximum error of the estimation Standard error of the estimate: the average error of the estimation.

How can we calculate the maximum error? Start from See:

2. In the case of sample means:

How can we calculate the value of k (1) it depends on the probability What do we know about the distribution of the sample means?

Distribution of sample means Size of sample Distribution Small Same as the popupulation Large (n>100) Normal distribution (Central limit theorem)

About the normal distribution X~N(E(X),  2 ) Special case E(X)=0,  2 = 1 transform into standard Normal distribution z~N(0,1) F(x)=  (z)  (-z)=1-  (z)

If x is a variable z is a Standardized variable Mean of z:0 Std. Dev of z:1

Apply for sample means

Calculation of value of k (2) 1-  given Pld.

Calculation of value of k (3) We should know the std. Dev of the population (  ) In the real life we know nothing about it 1. Instead of  we use s 2. Instead of normal distribution we use t-distribution!

t-distribution

Summary Examine Std dev.  or s? Small or large sample? In case of small samples can we assume the normal distribution? Type of sample (EV/FAE/R)?

Plan the sample size In real life the maximum error is given in advance. In this case what about the sample size?

Proportional stratified sample  known  unknown

With replacement Without replacement Estimate of proportion

Estimate of standard deviation