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

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 Population 1 2 3 4 5 6 7 8 9 10 Jack Susan Ann Bill Steve Sara Jane Julia Dave Ellen Draw random sample: n = 5

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 Susan Ann Sara Jane Julia Ellen 1 2 3 4 5 6 Females Jack Bill Steve Dave 1 2 3 4 Males Jack Susan Ann Bill Steve Sara Jane Julia Dave Ellen Proportion females = Proportion males =

Sampling from a Population Repeatedly draw random samples will differ from population different shape similar mean larger sample ---> closer to m ~

The Sampling Distribution of the Means Distribution of sample means from a single population Distribution of , not has m and s Find exact values take all possible samples or apply Central Limit Theorem ~

Notation Mean X sample m population mX 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. mX = m 3. Can calculate standard error of mean

Distributions: Variable vs Means f 70 85 100 115 130 IQ Score mean IQ Score

Standard Error of the Mean: Magnitude Desirable to have small sample means close m 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 m & standard error ~

How close is X to m? 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

Know/want Diagram: Sampling Distribution of Means Table: column A or B z score area under distribution Table: z column