Sampling Distributions & Standard Error

Lesson 7

Populations & Samples

Research goals Learn about population Characteristics that widely apply Impossible/impractical to directly study

Research methods Study representative sample Introduce sampling error ~

X

Sampling Error

Difference between sample

statistic and population parameter result of choosing random sample

Many potential samples With different ~

X

sandX

Sampling Distributions

Samples from a single population Repeatedly draw random samples Every possible combination

Calculate a test statistic (e.g., t test) One-sample: or Independent samples:

Results sampling distribution and ~

X

21 XX

The Distribution of Sample Means Distribution of means for many samples from

a single population Repeatedly draw random samples Calculate

Sampling variation (or sampling error) will differ from population different shape similar mean larger sample closer to~

sandX

Samples: n=10

#1 #2

#3 #4

Law of Large Numbers

Large sample size (n) give better estimates of parameters i.e., better fit

of estimatebetter a becomesX:

of estimatebetter a becomess:

estimate good ,30 if* n:

estimate good moderately ,6 if* n:

Parameters: Distribution of X

Results in narrower distribution Has and

Find exact values take all possible samples or apply Central Limit Theorem ~

Central Limit Theorem

1.

2.

nX

APA style: SE Also SEM ~

X ofon distributi of

X ofon distributi of

)(mean theoferror standard calledX

orn

sX

Central Limit Theorem 3. As sample size (n) increases

the sampling distribution of means approaches a normal distribution

even if parent population not normaldistribution of variable (or X)

Very Important! In n ≥ 6, then… probabilities from standard normal

distribution useful Because we study samples ~

Distributions: Xi vs X

1301008570

f

IQ Score

= 100 = 15n = 9

9

15

nM

5

11595 105

mean IQ Score90 110

Standard Error of the Mean: Magnitude

Small standard error better fit sample means close m More representative sample

Depends on n and large sample size & small little control can increase sample size

increase value of denominator ~

Using the distribution of X

Use samples to describe populations is it representative of population? how close is ?

Sample means normally distributed Use z table

find area under curve only slight difference in z formula ~

toX

Conducting an experiment

Same as randomly selecting...

nX

For a sample size n

with mean =

& standard error

XX ofon distributi from one

Calculating z scores

X

Xz

X means samplefor

X

z

Xscores raw for

How close is X to ?

means are normally distributed Use area under curve

between mean and 1 standard error above the mean

34% Same rules as any normal distribution

compute z score ~

Distribution of Sample Means is Normal

1 20-1-2

.34

.14

f

standard error of mean

.02

.34

.14

.02

)(X

z scores & Distribution of X

What are z scores that define boundaries of middle 95% of ? p in left & right tails = .025 + .025 Look up z scores Left tail = - 1.96; right tail = + 1.96

Boundaries for middle 99% of ? ~

X

X

Distribution of Sample Means is Normal

1 20-1-2

f

z scores )(X

Boundaries for middle 95% (or .95) of sample means?

-1.96 +1.96

for middle 99% (or .95) of sample means?

-2.58 +2.58

Using z scores

Sample Mean z score

area under curveor

proportionOr

probabilityor

percentage

Table: large/smaller portion column

Table: z column

X

Xz

X

zX

X