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 BA XX

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 ~

JackSusanAnnBillSteveSaraJaneJuliaDaveEllen

1 2 3 4 5 6 7 8 910

Simple Random Sampling

Draw random sample: n = 5

Population

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 ~

JackSusanAnnBillSteveSaraJaneJuliaDaveEllen

Stratified Random SamplingDraw random sample: n = 5

JackBillSteveDave

1 2 3 4

Males

SusanAnnSaraJaneJuliaEllen

1 2 3 4 5 6

FemalesPopulation

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 , not X

X

has and 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 population X 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

nX

Distributions: Variable vs Means

115 1301008570

f

IQ Score

= 100 = 15n = 9

X

mean IQ Score

Standard Error of the Mean: Magnitude

sample means close Depends on n and

large sample size & small little control can increase sample size

divide by larger number ~

Desirable to have small X

with mean & standard error ~

Sampling Distribution of the Means: Use

Conducting an experiment randomly selecting...

nX

one X for sample size n

from population of X

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

X

Xz

except use X and X

X

zX

Know/want Diagram: Sampling Distribution of Means

z score area under distribution

Table: column A or B

Table: z column

X

Xz

X

X

zX