Sampling Distributions: Understanding Parameter and Sample Statistic - Prof. Salvador Geza, Study notes of Data Analysis & Statistical Methods

Download Sampling Distributions: Understanding Parameter and Sample Statistic - Prof. Salvador Geza and more Study notes Data Analysis & Statistical Methods in PDF only on Docsity!

Ch

t^

Ch

apter

Sampling

Distributions

Parameter

vs.

Sample

Statistic

-^

A

parameter

is

a

numerical

descriptive

measure

of

a

population

Its

value

is

almost

always

unknown.

-^

A

sample

statistic

is

a

numerical

descriptive

measure

of

a

sample

.^

It

is

calculated

from

the

observations.

•^

In

practice,

sample

statistics

are

used

to

estimate population

parameters.

Parameter

Statistic

Mean V^

i

μ

x

ˆ μ

2

(^2) ˆ

2

Variance St.

DeviationProportion

2

(^2) ˆ

s

σ

2 σ

ˆ p

p

s

ˆ σ

σ

Properties

of

Sampling

Distributions

•^

If the

sampling

distribution

of

a

sample

statistic

has

a

mean

equal

to

the

population

parameter

the

statistic

is

intended

to

estimate,

the

statistic is said to be an

unbiased estimate

of the parameter

statistic

is

said

to

be

an

unbiased

estimate

of

the

parameter

•^

If two

alternative

sample

statistics

are

both

unbiased,

the

one

with

the smaller standard deviation is preferred (

minimum variance

the

smaller

standard

deviation

is

preferred

( minimum

variance

Sampling

Distribution

of



Properties

of

the

Sampling

Distribution

of

•^

The

mean

of the sampling

•^

The standard deviation of the

The

mean

of

the

sampling

distribution

equals

the

mean

of

the

population

The

standard

deviation

of

the

sampling

distribution

(the

standard

error

of

the

mean)

equals:

μ

μ

=

=

) ( x E

x^

n

x

σ

σ

=

n

( )

1

1

⎞ ⎛ ⎞

⎛

( ) ( )

n

x

V x V

n n x n E x E

x

i

2 2 2 2 1 1

1

1

σ

σ

σ

μ

μ μ

=

=

× ⎞⎟ ⎛⎜ = ⎞⎟

⎛⎜

=

=

⎞×⎟ ⎠ ⎛⎜ ⎝ = ⎞⎟ ⎠

⎛⎜ ⎝

∑ ∑

2

) (

i x^ i V

x E If

(^

)^

n n n x n V x V x

i

σ

σ

=

=

× ⎟ ⎠ ⎜ ⎝ = ⎟ ⎠

⎜ ⎝ =^

∑

Example

The

caffeine

content

(mg)

was

examined

for

a

random

sample

of

cups

of

black

coffee

dispensed

by

a

new

machine.

The

mean

and

standard deviation were 100 mg and 7.1 mg, respectively.standard

deviation

were

mg

and

mg,

respectively.

a)

Find

the

probability

that

the

average

caffeine

contents

will

be

b

d 101

b etween

and

mg.

b)

How

likely

are

you

to

get

on

those

cups

an

average

caffeine

content

higher

than

mg.

Central

Limit

Theorem

(CLT)

•^

The

sampling

distribution

of

,^

based

on

a

random

sample

of

n

observations,

will

be

approximately

normal,

i.e.

•^

Sampling distributions of averages will become more like a normal

) / , ( ~ ) , ( ~ n

N

x

iid

x i

σ μ

σ μ

≈≈>

•^

Sampling

distributions

of

averages

will

become

more

like

a

normal

distribution

as

n

increases

(regardless

of

the

shape

of

the

population

of

individual

measurements!!!).

Example

The

IMECA

is

an

index

used

to

determine

the

ozone

level

in

a

city.

Depending

upon

the

IMECA

reading,

it

might

not

be

safe

to

jog

or

even

to

go

outside.

Readings

in

the

‐^100

range

are

considered

‘normal’,

‐^200

are

‘unsatisfactory’,

‐^200

means

‘condition

bad’,

and

‐^500

means

‘conditions

very

bad’.

Suppose

that

an

industrial

region

has

an

average

IMECA

reading

of

with

a

g^

g^

g

standard

deviation

of

a) If a sample of 50 days is measured what is the samplinga)

If

a

sample

of

days

is

measured

,^ what

is

the

sampling

distribution

of

a

sample

mean?

b)

Find

the

probability

that

for

a

sample

of

days,

the

average

IMECA

reading

is

more

than

Example

a)

If

a

sample

of

days

is

measured,

what

is

the

sampling

distribution

of

a

sample

mean?

b) Find the probability that for a sample of 50 days the average IMECA

b)

Find

the

probability

that

for

a

sample

of

days

,^ the

average

IMECA

reading

is

less

than

⎛^ ⎜ ⎜ ⎜ ⎝

z

P

x

P

⎝^ z

P