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Sampling

Variability

and

Significance

Sampling

Variability

and

Significance

-^ Measures

of^ group

variability

from^

sample

to^ sample

or^ sample

to^ population. • Such^ a

study

involves

3 steps.

1.^ Selection

of^ sufficiently

large^

and^ random

samples

representative

of^ the

population

from^ which

they^ are

drawn.

2.^ Finding

the^ probability

or^ relative

frequency

of^ the

sample

results,

occurring

by^ chance.

3.^ Drawing

the^ inference,

if^ the^

probability

of^ a^ sample

value^

is

found^

higher

or^ lower

than^ the

probability

of^ its^

occurrence

by

chance. • In^ individual

variation,

SD(s)^

and^ shape

of^ normal

distribution

or^ curve

were^

found^

to^ be^ good

measures

of^ dispersion

of

observations

around

the^ mean

(x).^ Now

we^ shall

determine

if

the^ standard

error^

(standard

deviation

of^ means,

(x)^ and

shape

of^ sampling

distribution

or^ normal

curve

are^ good

measures

of

dispersion

of^ means

(X)^ around

the^ population

mean.

-^ If^ the

number

of^ samples

is^ large,

their

values

may^

be^ grouped

as^ done

in^ a^

frequency

distribution

table.

It^ will

be^ seen

that

the

samples

follow

a^ normal

distribution.

Normal^ distribution

of^ sample

values^ (means)

-^ In^ any

such

sampling

distribution:

1. Population

value

± 1^ SE

limits

Include

68%^

of^ the

sample

values

and^ fairly

large

number,

i.e.,^ 32%

samples

will^ have

higher

or^ lower

values

(16%

on

either

side).

2. Population

value

± 1.

SE^ limits

include

95%^

of^ the

sample

values.

Few,

only^

5%^ sample

values,

will^ fall

beyond

this^ range

or^ these

limits.

In^ other

words.

chances

of^ such

high^

or^ low

values

being

normal

will

be^ 5%.3. Range

defined

by^ population

value

± 2.

SE

Includes

99%^

of^ the

estimates,

hence,

an^ estimate

higher

or^ lower

than

that^

will^ be

obtained

by^ chance

in^ 1%

cases.

i.e.,^ very

rarely.

Such

high^

or^ low

value

will^ probably

be^ due

to^ some

factor(s)

SIGNIFICANCE

-^ After

making

experiments

in^ medical

problems,

certain

results

like^ means

and

proportions

are^ obtained

which

vary

from

sample

to^ sample

and^

sample

to^ universe.

-^ Next

is^ the

stage

of^ Interpretation

of^ results

or

drawing

statistical

Inferences

or^ conclusions.

-^ In^ other

words,

the^ observer

or^ experimenter

wants

to^ know

the^ significance

of^ the

difference

he^ has

observed

in^ his

result

as^ compared

with

that^

of^ the

population

or^ with

that^

of^ another

worker,

e.g.,

he^ finds

that^

mean

blood

pressure

of^ his

sample

is^ higher

than

that

observed

by

another

worker

or^ another

worker

may

find^

the

cure^

rate^ with

chloramphenicol to

be^ higher

than^

with^

tetracycline

and^

so^ on.

Estimation

of^ Population

Parameter

-^ We

cannot

draw

large

number

of^ samples

covering

the^ entire

population

in^ order

to^ find

the^ population

parameter

().^

So,^ we

calculate

the^ same

from

a^ sample

statistic

such

as

-^ We

then

set^ up

certain

limits

on^ both

sides

of

the^ population

mean

()^ on

the^ basis

of^ the

fact^ that

means

of^

samples

of^ size

30 or

more

are^ normally

distributed

around

the

population

mean

-^ These

limits

are^ called

the^ confidence

limits

and

the^ range

between

the^ two

is^ called

the

confidence

interval

-^ As^

per^ normal

distribution

of^ samples,

we^ say

with^

confidence

or^ we

are^ sure

that^

95%^

of^ the

sample

means

will

lie^ within

the^ confidence

limits

of

-^ 95%

confidence

interval

thus

obtained

will

contain

of^ sample

means.

-^ Conversely,

population

mean

()^ will

also^

fall

within

these

confidence

limits

or^ lie

in^ the

confidence

intervals

between

at^ 95%

confidence

interval.

The^ range,

thus

obtained,

will^ contain

population mean

in^ 95%

cases.

-^ This

also^

Implies

that^

any^ sample

or^ universe

value

lying

outside

the^ range

mean

SE

will^ be

rare.

-^ The

probability

or^ relative

frequency

of^ such

occurrence

by^ chance

will^ be

5%^ or

out^ of

one,^

i.e.,^ once

in^20

times.

-^ If^ confidence

limits

are^ extended

to^ cover

wider

interval

or^ range

between

mean

SE,^ we

can^ say

with

confidence

that

other

sample

means

as^ well

as^ that

of^ universe

or^ population

would

fall^ in

these

limits.

-^ The

limit

of^ the

region

at^ which

we^ no

longer

regard

the^ chance

to^ be

operating

Is^ called

the

level

of^ significance

.^ It^ separates

the^ shaded

areas

in^ one

or^ two

tails^

of^ the

area

under

the

normal

curve

form

the^ plain

area.

Shaded^

areas^ indicate

the^ level

of^ significance

lying^ at

one^ end

only

-^ If^ the

chance

limit

is^ set

at^ mean

SE,^ it

implies

5%^ or

level

of^ significance,

also

called

the^ critical

level

of^ significance.

-^ A^ value

lying

beyond

this^

area^

is^ said

to^ be

significantly

different

from

the^ population

value.

-^ At^

this^ level,

such

extreme

values

will^ occur

by

chance

only

5 times

In^100

experiments.

Shaded^ areas

indicate the^ level

of^ significance lying^ at^ one

end^ only docsity.com

Testing

of^ Statistical

Hypothesis

-^ To^ test

statistical

hypotheses

about

the^ population

parameter

or^ true

value^

of^ universe,

two^ hypotheses

or

presumptions

are^ made

to^ draw

the^ inference

from^

the

sample

value.

1. A^ null

hypothesis

or^ hypothesis

of^ no^

difference

(H^ )o^

between

statistic

of^ a^ sample

and^ parameter

of^ population

or^ between

statistic

of^ two

samples.

This^ hypothesis

nullifies

the^ claim

that^ the

experimental

result

is^ different

from^ or

better

than^ the

one^ observed

already.

2. The

alternative

hypothesis

of^ significant

difference

(H^ )^1

stating

that^ the

sample

result

is^ different—greater

or

smaller

than^ the

hypothetical

value^

of^ population,

e.g.

weight

gain^ or

loss^ due

to^ new

feeding

regimen.

-^ By^

this^ we

shall

adopt

a^ procedure

to^ choose

between

null^

hypothesis

(H^ )^ o^

and^ alternate

hypothesis

(H^ )^1

by^ applying

relevant

statistical

technique. • A^ test

of^ significance

such

as^ Z‐

test^ is

performed

to^ accept

the^ null

hypothesis

Horo^

to^ reject

it

and^ accept

the^ alter

native

hypothesis

H.^1

-^ To^

make

minimum

error

in^ rejection

or

acceptance

of^ H

,^ weo divide

the^ sampling

distribution

or^ the

area

under

the^ normal

curve

into^ two

regions

or^ zones.

I.^ A

zone

of^ acceptance II.^ A

zone

of^ rejection