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Lecture 7 How certain are we? Sampling and the normal distribution Chapter 11 – 1 Lecture 6: Summary • If we take a simple random sample – from a well-defined population • we expect – that the sample mean – is “probably” “close” to the population mean • By “close” we mean “within ~2 standard errors” Lecture 7: Preview • Today, we’ll learn that “probably” means – in 95% of all samples Chapter 11 – 2 Overview • Review of sampling distributions • Sampling distributions have a “normal” shape • Properties of the “normal” distribution, e.g.: – In 95% of all samples, • the sample mean • is within 1.96 standard errors • of the population mean Chapter 11 – 3 Repeated sampling Population: All US households mY=1.75 Y Y Y Y Y YY Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y All possible samples Y Y Y Y Y Y Y Y N=4 Y 1.25 N=4 Y 2.50 … Each Y represents the number of children in a household Chapter 11 – 4 Notation Mnemonics: Population measures are called Parameters. Sample measures are called Statistics. The P words and S words go together. Population parameters use Greek letters Sample statistics use Roman letters m=Greek m p=Greek p s=Greek s The population is the source of the sample. Greek culture was the source of Roman culture. Chapter 11 – 5 Population Population Variable population mean population standard deviation US households Y (# of children) mY=1.75 sY=1.62 Chapter 11 – 6 Sample CHILDS 1 0 2 2 sample mean 1.25 Within sample… Sample size Variable sample mean sample standard deviation N=4 Y (# of children) Y 1.25 sY=.92 Chapter 11 – 7 Population of samples CHILDS 1 0 2 2 sample mean 1.25 Y CHILDS 2 4 0 4 sample mean 2.50 Y Mean Standard error # of samples: infinite but just N=4 adults per sample Across samples… “Variable” … Y mY mY here 1.75 sY sY / N here 1.62 / 41/2 = 1.62 / 2 = 0.81 (Std. dev. of sample means) Chapter 11 – 8 As sample size (N) grows… …standard error shrinks! …shape of sampling distribution gets closer to “normal”! 1.75 .81 1.75 .405 1.75 .2025 Chapter 11 – 9 Normal distribution • symmetric • bell-shaped • very specific numeric properties Chapter 11 – 10 “Margin of error” In your course binder, find the “z (standard normal)…table”. Look for this line. Confidence z 94% 1.88 95% 1.96 96% 2.05 This means: In 95% of all samples, the sample mean is within 1.96 standard errors of the population mean. +/- 1.96 (or 2) standard errors often called “margin of error”Chapter 11 – 11 Example 1 Again “population”: US adults Variable: Y: “How many children have you ever had?” mY=1.75, sY=1.62. Consider samples of size N=16. 95% of all sample means Y are within 1.96 standard errors of pop. mean —i.e., in 95% mY Zs Y mY 1.96s Y mY 1.96s Y / N 1.75 1.96(1.62 / 16 ) 1.75 1.96(1.62 / 4) 1.75 .79 0.96 to 2.54 Chapter 11 – 12 More on sampling error Look for this line. Confidence z 98% 2.33 99% 2.58 99.9% 3.29 This means: In 99% of all samples, the sample mean is within 2.58 standard errors of the population mean. (1% of samples have means that are further away.) Chapter 11 – 13 Example 2 Variable: Y: “How many children have you ever had?” mY=1.75, sY=1.62. Consider samples of size N=45. 99% of all sample means are within 2.58 standard errors of the population mean —i.e., in 99% mY 2.58s Y mY 2.58s Y / N 1.75 2.58(1.62 / 45 ) 1.75 .62 1.13 to 2.37 Chapter 11 – 14 Sampling error: Exercise Complete the following: 90% of all samples have means within _______ SE’s of the population mean. Complete the following: If researchers take samples of 100 US adults, 90% of the time the sample will average between _______ and _________ children. Chapter 11 – 15 Summary: Central Limit Theorem (CLT) • The sampling distribution of Y – has mean mY mY – and standard error s Y sY / N • As the sample size N gets larger, – the standard error gets smaller – and the sampling distribution gets closer to “normal.” • So – larger samples give • closer • more predictable – approximations to the population mean Chapter 11 – 16 Summary • Lecture 6 (Law of Large Samples) – If we take simple random samples • from a well-defined population – we expect • that the sample means • is “usually” “close” to the population mean • Lecture 7 (Central Limit Theorem) – If by “close” • we mean “within 1.96 standard errors” – then by “usually” • we mean “in 95% of all samples” – For other definitions of “close” and “usually,” • see the “z (standard normal)…table” in your course binder Chapter 11 – 17 Teaser: Lecture 8 (Confidence intervals) • So if we take – just one sample • we can guess – that the population statistic is “close” • and we’ll “usually” be right Chapter 11 – 18