1. Samples & the Sampling Distributions of the Means
Chapter 7
Homework: 1 (a-i), 2-8
sketch: use mean & standard deviation or
mean & standard error

2. Sampling Goal of sampling: describe population
Sample: subset of population
Could take many different samples
error introduced

3. Sampling Want representative sample
members reflect characteristics of the population
not extremes
best chance for representative...
choose members at random ~

4. 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 ~

5. 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 ~

6. 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 ~

7. 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

9. 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 ~

10. 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 =

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

13. 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 ~

14. 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 ~

15. 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

16. Distributions: Variable vs Meansf 70 85
100
115
130
IQ Score
mean IQ Score

17. 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 ~

18. 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 ~

19. 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 ~

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

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