© 2014 by Pearson Higher Education, Inc Upper Saddle River, New Jersey All Rights Reserved HLTH 300 Biostatistics for Public Health Practice, Raul Cruz-Cano, Ph.D. 3/31/2014, Spring 2014 Fox/Levin/Forde, Elementary Statistics in Social Research, 12e Chapter 6: Samples and Populations 1

© 2014 by Pearson Higher Education, Inc Upper Saddle River, New Jersey All Rights Reserved Distinguish between populations and samples Describe the methods of random and nonrandom sampling Understand the concept of sampling error Understand the characteristics of the sampling distribution of means Calculate confidence intervals CHAPTER OBJECTIVES

Distinguish between populations and samples Learning Objectives After this lecture, you should be able to complete the following Learning Outcomes 6.1

4 Population (or universe) – a group of a set of individuals that share at least one characteristic Sample – a small number of individuals from the population Social researchers generally are not able to measure an entire population Limited by time and resources Sampling allows researchers to generalize Populations and Samples Population sample

Describe the methods of random and nonrandom sampling Learning Objectives After this lecture, you should be able to complete the following Learning Outcomes 6.2

Sampling 6.2 vs. Every member of the population has the same chance of being included Every member of the population does not have the same chance of being included

Nonrandom Sampling 6.2 Accidental or Convenience The sample is based on what is convenient for the researcher Accidental or Convenience The sample is based on what is convenient for the researcher Quota The sample is drawn in proportion to the population Quota The sample is drawn in proportion to the population Judgment or Purposive The sample is drawn according to logic, common sense, or judgment Judgment or Purposive The sample is drawn according to logic, common sense, or judgment

Random Sampling 6.2 Simple Random Similar to drawing numbers from a hat Simple Random Similar to drawing numbers from a hat Systematic Every n th member is included Systematic Every n th member is included Stratified Divides the population into homogenous subgroups and then samples Stratified Divides the population into homogenous subgroups and then samples Cluster or Multistage Samples are drawn at different levels Cluster or Multistage Samples are drawn at different levels

Understand the concept of sampling error Learning Objectives After this lecture, you should be able to complete the following Learning Outcomes 6.3

10 By chance alone, we can always expect some difference between a sample and the population from which it is drawn We use different symbols for samples as compared to populations Sampling Error MeasurePopulationSample Mean Standard Deviation Standard Error

Understand the characteristics of the sampling distribution of means Learning Objectives After this lecture, you should be able to complete the following Learning Outcomes 6.4

Sampling Distribution of Means 6.4 The sampling distribution of means approximates a normal curve The mean of the sample distribution of means (the mean of means) is equal to the true population mean The standard deviation of a sampling distribution of means is smaller than the standard deviation of the population A frequency distribution of a large number of random sample means that have been drawn from the same population Characteristics of the Sampling Distribution of Means:

6.4 Figure 6.3

6.4 Figure 6.4 If you have enough data then this is true regardless of the distribution of X!

Calculate confidence intervals Learning Objectives After this lecture, you should be able to complete the following Learning Outcomes 6.5

Figure 6.6

The standard deviation of the sampling distribution of means Standard Error of the Mean

The range of mean values within which the population mean is likely to fall Confidence intervals can be calculated at different levels: Confidence Intervals How sure do you want to be?

Figure 6.8

Rarely do we actually know the standard deviation of the population This makes it impossible to calculate the standard error of the mean But we can estimate it: The t Distribution

When we use the sample standard deviation (and not the population standard deviation) the distribution is not normal –We have to use the t distribution instead of the z distribution –The value of t can be found in Table C The t Distribution Continued

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6.5 Figure 6.9

24 Small Sample Size Example Let’s find CI for data in Problem 12 Basically, we just follow the steps in pages )Find sample mean from raw data (Chapter 3) 2)Calculate standard deviation of the sample from raw data (Chapter 4) 3)Estimate standard error of the mean: divide 2) by SQRT(N-1) 4)Determine t (use Table C in page 552) 5)Obtain Margin of Error: Multiply 3) by 4) 6)Add and subtract 5) from 1)

25 Large Sample Size Example Let’s solve Problem 21 We just follow the steps in pages (We will concentrate in the cases where sample mean and standard deviation are given, hence we begin in Step 3) 1)Find sample mean from frequency table(Chapter 3) 2)Calculate standard deviation of the sample from a frequency table (Chapter 4) 3)Estimate standard error of the mean: divide 2) by SQRT(N-1) 4)Determine t (use closest of the last rows of Table C in page 552) 5)Obtain Margin of Error: Multiply 3) by 4) 6)Add and subtract 5) from 1)

The same rationality is used to estimate population proportions The only difference is that we use the z distribution as opposed to the t distribution used to estimate population means Estimating Proportions In other words, you just need the proportion to find the sample mean and standard error of the proportion

27 Proportion Examples Let’s solve Problem 34! We just follow the steps in pages )Estimate standard error of the mean (previous slide = 1 formula in page 207) 2)Determine t (use last row of Table C in page 552) 3)Obtain Margin of Error: Multiply 1) by 2) 4)Add and subtract 3) from the given p

28 More Examples Problem 9, 26 and 29

29 CI’s can be used for hypothesis testing for a single variable vs. a constant

30 HW#6 Problem 13, 27 and 30

© 2014 by Pearson Higher Education, Inc Upper Saddle River, New Jersey All Rights Reserved Samples are drawn from populations and can be used to generalize findings Several forms of random and nonrandom sampling methods are used in the social sciences Due to random chance, the sample will differ from the population. This is referred to as sampling error The sampling distribution of means has several important characteristics that allow researchers to generalize their findings The specific method used to calculate a confidence interval is determined by whether the researcher knows the population standard deviation and whether means or proportions are being estimated CHAPTER SUMMARY