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Samples & the Sampling Distributions of the Means Chapter 7 Homework: 1 (a-i), 2-8 sketch: use mean & standard deviation or mean & standard error Sampling Goal of sampling: describe population Sample: subset of population Could take many different samples error introduced X XA XB Sampling Want representative sample members reflect characteristics of the population not extremes best chance for representative... choose members at random ~ Random Samples Each member has equal chance of being selected for sample independent of selection of other members Helps avoid biases of experimenters Focus: simple & stratified random sampling also other methods ~ Simple Random Sampling All members of population treated equally regardless of characteristics e.g., bag of M&Ms Set of random digits better computer-generated table of random digits Table A.10 ~ Simple Random Sampling: Procedure 1. Assign # to each population member 2. Go to random digit table 3. Quasi-randomly select “seed” 4. Start with seed, read # digits required e.g., N=20, use 2 digits Read L --> R, ignore spaces If # already used or not in range discard & got to next ~ Simple Random Sampling 1 2 3 4 5 6 7 8 9 10 Population Jack Susan Ann Bill Steve Sara Jane Julia Dave Ellen Draw random sample: n = 5 Table A.10 2000 Random digits 1 6 11 16 21 31 36 1 10480 15011 01536 02011 81647 91646 69179 2 22368 46573 25595 85393 30995 89198 27982 3 24130 48360 22527 97265 76393 64809 15179 4 42167 93093 06243 61680 07856 16376 39440 5 37570 39975 81837 16656 06121 91782 60468 Stratified Random Sampling If population has subgroups of interest representative sample has same proportion of subgroups Number subjects within each group females: 1, 2, 3, 4, .... males: 1, 2, 3, 4, ... Use same procedures as simple random sampling new seed for each group ~ Stratified Random Sampling Draw random sample: n = 5 Population Jack Susan Ann Bill Steve Sara Jane Julia Dave Ellen 1 2 3 4 5 6 Females Susan Ann Sara Jane Julia Ellen 1 2 3 4 Males Jack Bill Steve Dave Proportion females = Proportion males = Sampling from a Population Repeatedly draw random samples will differ from population different shape similar mean larger sample ---> closer to ~ The Sampling Distribution of the Means Distribution of sample means from a single population Distribution of X , not X has and s Find exact values take all possible samples or apply Central Limit Theorem ~ Notation Mean X sample population X population of sample means Standard deviation s sample s population sX population of sample means standard error of the mean ~ Central Limit Theorem Describes sampling distribution of mean Specifies shape, center, width 1. It is a normal distribution even if parent population not normal if n > 30 2. X = 3. Can calculate standard error of mean s X s n Distributions: Variable vs Means = 100 s = 15 n=9 f s 70 85 IQ Score 100 115 X 130 mean IQ Score Standard Error of the Mean: Magnitude Desirable to have small s X sample means close Depends on n and s large sample size & small s little control s can increase sample size divide by larger number ~ Sampling Distribution of the Means: Use Conducting an experiment randomly selecting... one X for sample size n from population of X with mean & standard error s X s n ~ How close is X to ? means are normally distributed Use area under curve between mean and 1 standard error from the mean .34 Same rules as any normal distribution compute z score ~ Z scores, X & proportions Calculate just like values of X except use X and s X z X sX X zs X Know/want Diagram: Sampling Distribution of Means X zs X Table: column A or B z score X z X sX area under distribution Table: z column

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