Samples & the Sampling Distributions of the Means

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 ~

Females Males 1 2 3 4 Susan Ann Sara Jane Julia Ellen Jack Bill Steve Dave 1 2 3 4 5 6 Proportion females = Proportion males = Stratified Random Sampling Draw random sample: n = 5 Population Jack Susan Ann Bill Steve Sara Jane Julia Dave Ellen

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

Distribution of , not The Sampling Distribution of the Means • Distribution of sample means • from a single population • 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 m = 100 s= 15 n = 9 f 70 85 100 115 130 IQ Score mean IQ Score

Desirable to have small Standard Error of the Mean: Magnitude • 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 ~

one X • from population of X Sampling Distribution of the Means: Use • Conducting an experiment • randomly selecting... • for sample size n • 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 ~

except use X and s X Z scores, X & proportions • Calculate just like values of X

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

Samples & the Sampling Distributions of the Means