Chapter 11 – 1 Lecture 7 How certain are we? Sampling and the normal distribution

Chapter 11 – 2 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 – 3 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 – 4 Repeated sampling 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 Each Y represents the number of children in a household Population: All US households Y … All possible samples N=4  Y =1.75

Chapter 11 – 5 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  =Greek m  =Greek p  =Greek s The population is the source of the sample. Greek culture was the source of Roman culture.

Chapter 11 – 6 Population US households VariableY (# of children) population mean  Y =1.75 population standard deviation  Y =1.62

Chapter 11 – 7 Sample Sample sizeN=4 Variable Y (# of children) sample mean sample standard deviation s Y =.92 Within sample…

Chapter 11 – 8 Population of samples … # of samples: infinite “Variable” Mean Standard error here 1.75 here 1.62 / 4 1/2 = 1.62 / 2 = 0.81 (Std. dev. of sample means) Across samples… but just N=4 adults per sample

Chapter 11 – 9 As sample size (N) grows… …standard error shrinks! …shape of sampling distribution gets closer to “normal”!

Chapter 11 – 10 Normal distribution symmetric bell-shaped very specific numeric properties

Chapter 11 – 11 “Margin of error” This means: In 95% of all samples, the sample mean is within 1.96 standard errors of the population mean. +/ (or 2) standard errors often called “margin of error” In your course binder, find the “z (standard normal)…table”. Look for this line.

Chapter 11 – 12 Example 1 Again “population”: US adults Variable: Y: “How many children have you ever had?”  Y =1.75,  Y =1.62. Consider samples of size N=16. 95% of all sample means are within 1.96 standard errors of pop. mean —i.e., in 95%

Chapter 11 – 13 More on sampling error 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.) Look for this line.

Chapter 11 – 14 Example 2 Variable: Y: “How many children have you ever had?”  Y =1.75,  Y =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%

Chapter 11 – 15 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 – 16 The sampling distribution of –has mean –and standard error 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 Summary: Central Limit Theorem (CLT)

Chapter 11 – 17 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 – 18 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