Sampling Design and Analysis MTH 494 LECTURE-11 Ossam Chohan Assistant Professor CIIT Abbottabad.

Slides:



Advertisements
Similar presentations
SADC Course in Statistics Estimating population characteristics with simple random sampling (Session 06)
Advertisements

Introduction Simple Random Sampling Stratified Random Sampling
Estimating a Population Proportion
CmpE 104 SOFTWARE STATISTICAL TOOLS & METHODS MEASURING & ESTIMATING SOFTWARE SIZE AND RESOURCE & SCHEDULE ESTIMATING.
© 2011 Pearson Education, Inc
Estimation in Sampling
1 Virtual COMSATS Inferential Statistics Lecture-7 Ossam Chohan Assistant Professor CIIT Abbottabad.
Chapter 10: Estimating with Confidence
Economics 105: Statistics Review #1 due next Tuesday in class Go over GH 8 No GH’s due until next Thur! GH 9 and 10 due next Thur. Do go to lab this week.
QBM117 Business Statistics Statistical Inference Sampling 1.
Dr. Chris L. S. Coryn Spring 2012
Point and Confidence Interval Estimation of a Population Proportion, p
Why sample? Diversity in populations Practicality and cost.
Fundamentals of Sampling Method
STAT262: Lecture 5 (Ratio estimation)
1 1 Slide © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole.
QMS 6351 Statistics and Research Methods Chapter 7 Sampling and Sampling Distributions Prof. Vera Adamchik.
7/2/2015 (c) 2001, Ron S. Kenett, Ph.D.1 Sampling for Estimation Instructor: Ron S. Kenett Course Website:
A new sampling method: stratified sampling
8-1 Introduction In the previous chapter we illustrated how a parameter can be estimated from sample data. However, it is important to understand how.
Formalizing the Concepts: Simple Random Sampling.
Understanding sample survey data
Sampling Designs Avery and Burkhart, Chapter 3 Source: J. Hollenbeck.
Scot Exec Course Nov/Dec 04 Ambitious title? Confidence intervals, design effects and significance tests for surveys. How to calculate sample numbers when.
Chapter 7 Sampling and Sampling Distributions n Simple Random Sampling n Point Estimation n Introduction to Sampling Distributions n Sampling Distribution.
Chapter 7 Estimation: Single Population
1 1 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole.
1 1 Slide © 2005 Thomson/South-Western Slides Prepared by JOHN S. LOUCKS St. Edward’s University Slides Prepared by JOHN S. LOUCKS St. Edward’s University.
Sampling: Theory and Methods
Chapter 7 Sampling and Sampling Distributions Sampling Distribution of Sampling Distribution of Introduction to Sampling Distributions Introduction to.
Virtual COMSATS Inferential Statistics Lecture-6
8.2 - Estimating a Population Proportion
Estimation and Confidence Intervals Chapter 9 McGraw-Hill/Irwin Copyright © 2012 by The McGraw-Hill Companies, Inc. All rights reserved.
1 1 Slide Chapter 7 (b) – Point Estimation and Sampling Distributions Point estimation is a form of statistical inference. Point estimation is a form of.
1 1 Slide © 2008 Thomson South-Western. All Rights Reserved Slides by JOHN LOUCKS St. Edward’s University.
Lecture 14 Sections 7.1 – 7.2 Objectives:
Chap 20-1 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chapter 20 Sampling: Additional Topics in Sampling Statistics for Business.
Population All members of a set which have a given characteristic. Population Data Data associated with a certain population. Population Parameter A measure.
PROBABILITY (6MTCOAE205) Chapter 6 Estimation. Confidence Intervals Contents of this chapter: Confidence Intervals for the Population Mean, μ when Population.
PARAMETRIC STATISTICAL INFERENCE
Section 8.1 Estimating  When  is Known In this section, we develop techniques for estimating the population mean μ using sample data. We assume that.
Sampling Design and Analysis MTH 494 Ossam Chohan Assistant Professor CIIT Abbottabad.
1 1 Slide Sampling and Sampling Distributions Sampling Distribution of Sampling Distribution of Introduction to Sampling Distributions Introduction to.
Basic Sampling & Review of Statistics. Basic Sampling What is a sample?  Selection of a subset of elements from a larger group of objects Why use a sample?
Chapter 18 Additional Topics in Sampling ©. Steps in Sampling Study Step 1: Information Required? Step 2: Relevant Population? Step 3: Sample Selection?
1 Chapter 7 Sampling and Sampling Distributions Simple Random Sampling Point Estimation Introduction to Sampling Distributions Sampling Distribution of.
LECTURE 3 SAMPLING THEORY EPSY 640 Texas A&M University.
Sampling Design and Analysis MTH 494 Lecture-30 Ossam Chohan Assistant Professor CIIT Abbottabad.
STA Lecture 181 STA 291 Lecture 18 Exam II Next Tuesday 5-7pm Memorial Hall (Same place) Makeup Exam 7:15pm – 9:15pm Location TBA.
Sampling Design and Analysis MTH 494 Ossam Chohan Assistant Professor CIIT Abbottabad.
Sampling Design and Analysis MTH 494 LECTURE-12 Ossam Chohan Assistant Professor CIIT Abbottabad.
1 Chapter 7 Sampling Distributions. 2 Chapter Outline  Selecting A Sample  Point Estimation  Introduction to Sampling Distributions  Sampling Distribution.
Section 10.1 Confidence Intervals
Understanding Sampling
Statistics for Decision Making Basic Inference QM Fall 2003 Instructor: John Seydel, Ph.D.
Sampling Design and Analysis MTH 494 Lecture-22 Ossam Chohan Assistant Professor CIIT Abbottabad.
SAMPLE SIZE.
Sampling Design and Analysis MTH 494 Lecture-21 Ossam Chohan Assistant Professor CIIT Abbottabad.
The inference and accuracy We learned how to estimate the probability that the percentage of some subjects in the sample would be in a given interval by.
Estimation and Confidence Intervals. Sampling and Estimates Why Use Sampling? 1. To contact the entire population is too time consuming. 2. The cost of.
Variability. The differences between individuals in a population Measured by calculations such as Standard Error, Confidence Interval and Sampling Error.
Estimation and Confidence Intervals
Variability.
Chapter 6 Inferences Based on a Single Sample: Estimation with Confidence Intervals Slides for Optional Sections Section 7.5 Finite Population Correction.
Chapter 4. Inference about Process Quality
Virtual COMSATS Inferential Statistics Lecture-11
Slides by JOHN LOUCKS St. Edward’s University.
CONCEPTS OF ESTIMATION
2. Stratified Random Sampling.
Chapter 7 Sampling and Sampling Distributions
Presentation transcript:

Sampling Design and Analysis MTH 494 LECTURE-11 Ossam Chohan Assistant Professor CIIT Abbottabad

Review 2

Confidence Intervals Confidence Interval: An interval of values computed from the sample, that is almost sure to cover the true population value. We make confidence intervals using values computed from the sample, not the known values from the population Interpretation: In 95% of the samples we take, the true population proportion (or mean) will be in the interval. This is also the same as saying we are 95% confident that the true population proportion (or mean) will be in the interval

Sample Size Estimation An investigator might have number of goals while handling samplings issue. Deciding amount of sampling error and balance the precision of estimates with the cost of survey especially in SRS. Estimating a sample size is one of the major goal of surveys. 4

Steps involved in sample size estimation Step-1 – What is expected of the sample and how much precision do I need? – What are the consequences of sample results. – How much error is tolerable. – A preliminary investigation, however, often needs less precision than an ongoing survey. 5

A wrong approach – Many people usually ask “what percentage of the population should I include in my sample”. – Ideally focus should be on precision of estimates. Precision is obtained through the absolute size of the sample, not the proportion of the population covered (except in very small populations) 6

Step-2 – Find an equation relating the sample size n and your expectations of the sample Step-3 – Estimate any unknown quantities and solve for n. Step-4 – If you feel that sample size (estimated) is too large to handle, go back and adjust your expectation and then try again. – Still if your sample size is large, then do think again to initiate your study. 7

Specify the Tolerable Error How much precision is needed (decided by investigator only). The desired precision is often expressed in absolute term, as Pr(|Estimator-Parameter|≤e)=1-α Where e is called Margin of Error Reasonable values for α and e must be decided by investigator. For many surveys of people in which a proportion is measured, e=0.03 and α=0.05 8

Sometimes you would like to achieve a desired relation precision, controlling coefficient of variation (CV) rather than the absolute Error. In that case, if Parameter ≠ 0, the precision may be expressed as 9

Find an Equation The simplest equation relating the precision and sample size comes from the confidence intervals in the previous section. To obtains absolute precision e, find a value of n that satisfies To solve this equation for n, we first find the sample n 0 that we would use for an SRSWR, that is 10

Then the desired sample size is Of course, if n 0 ≥N, we simply take a census with n=N 11

Example Suppose we want to estimate the proportion of recipes in the Better Homes and Gardens New Cook Book that do not involve animal products. We plan to take an SRS of the N=1251 test kitchen-tested recipes, and want to use a 95% CI with margin of error then n 0 = 12

Randomization theory results for Simple Random Sampling In this section we will prove some unbiased estimators. No distribution assumption are made about the y i ’s in order to ascertain that is unbiased for estimating population mean µ, like normality assumptions. Let us see how the randomization theory works for deriving properties of the sample mean in SRS. As in Cornfield ( 1944) _____________________________ 13

14

15

16

17

18

19

When should a Simple Random Sample Be Used? Avoid SRS in such situations – Before taking an SRS, you should consider whether a survey sample is the best method for studying your research question. – You may not have a list of the observation units, or it may be expensive in terms of travel time to take an SRS. – You may have additional information that can be used to design a more cost effective sampling scheme. 20

SRS should be used in following situations Little extra information is available that can be used when designing the survey like sampling frame. Person using the data insist on using SRS formula, whether they are appropriate or not. The primary interest is in multivariate relationships such as regression equations that hold for the whole population, and there are no compelling reasons to take a stratified or cluster sample. 21

Key Terms Used in SRS Unit Cluster sample: A probability sample in which each population unit belongs to a group, or cluster, and the clusters are sampled according to the sampling design. Confidence interval (CI): An interval estimate for a population quantity, for which the probability that the random interval contains the true value of the population quantity is known. Design-based inference: Inference for finite population characteristics based on the survey design, also called randomization inference. Finite population correction (fpc): A correction factor which, when multiplied by the with-replacement variance, gives the without-replacement variance. For an SRS of size n from a population of size N, the fpc is 1 − n/N. 22

Inclusion probability: π i = probability that unit i is included in the sample. Margin of error: Half of the width of a 95% CI. Model-based inference: Inference for finite population characteristics based on a model for the population, also called prediction inference. Probability sampling: Method of sampling in which every subset of the population has a known probability of being included in the sample. Sampling distribution: The probability distribution of a statistic generated by the sampling design. 23

Sampling weight: Reciprocal of the inclusion probability; w i =1/π i. Self-weighting sample: A sample in which all probabilities of inclusion π i are equal, so that all sampling weights wi are the same. Simple random sample with replacement (SRSWR): A probability sample in which the first unit is selected from the population with probability 1/N; then the unit is replaced and the second unit is selected from the set of N units with probability 1/N, and so on until n units are selected. 24

Simple random sample without replacement (SRS): An SRS of size n is a probability sample in which any possible subset of n units from the population has the same probability = (n!(N −n)!/N!) of being the sample selected. Standard error (SE): The square root of the estimated variance of a statistic. Stratified sample: A probability sample in which population units are partitioned into strata, and then a probability sample of units is taken from each stratum. Systematic sample: A probability sample in which every k th unit in the population is selected to be in the sample, starting with a randomly chosen value R. Systematic sampling is a special case of cluster sampling. 25