1 Univariate Inferences about a Mean Shyh-Kang Jeng Department of Electrical Engineering/ Graduate Institute of Communication/ Graduate Institute of Networking.

11

Univariate Inferences Univariate Inferences about a Meanabout a Mean

Shyh-Kang JengShyh-Kang JengDepartment of Electrical Engineering/Department of Electrical Engineering/

Graduate Institute of Communication/Graduate Institute of Communication/

Graduate Institute of Networking and Graduate Institute of Networking and MultimediaMultimedia

ScenariosTo test if the following statements are plausible– A clam by a cram school that their

course can increase the IQ of your children

– A diuretic is effective– An MP3 compressor is with higher

quality– A claim by a lady that she can

distinguish whether the milk is added before making milk tea

22

33

Evaluating Normality of Univariate Evaluating Normality of Univariate Marginal DistributionsMarginal Distributions

sticcharacterith for theon distributi

normal assumedan from departure indicate

628.0)046.0)(954.0(3954.0ˆor

396.1)317.0)(683.0(3683.0ˆeither

)22(in lying data ofportion :ˆ

)(in lying data ofportion :ˆ

data, ofsymmetry checkingAfter

2

1

2

1

i

nnp

nnp

sx,sxp

sx,sxp

33

Tests of HypothesesDeveloped by Fisher, Pearson, Neyman, etc.Two-sided

One-sided)hypothesis ve(alternati:

)hypothesis (null:

01

00

H

H

)hypothesis ve(alternati:

)hypothesis (null:

01

00

H

H

44

Assumption under Null Hypothesis

)1,0(:/

)/,(:

),(:

0

20

20

Nn

XZ

nNX

NX

55

Rejection or Acceptance of Null Hypothesis

66

Student’s t-Statistics

n

ii

n

ii

XXn

s

Xn

Xns

Xt

1

22

1

0

1

1

1,

/

77

88

Student’s Student’s tt-distribution-distribution

2

12

2

1)

2(

)2

1(

)(

/

f

f

tf

f

f

tf

f

Zt

Student’s Student’s tt-distribution-distribution

99

1010

Student’s Student’s tt-distribution-distribution

Origin of the Name “Student”

Pseudonym of William Gossett at Guinness Brewery in Dublin around the turn of the 20th CenturyGossett use pseudonym because all Guinness Brewery employees were forbidden to publishToo bad Guinness doesn’t run universities

1111

1212

Test of HypothesisTest of Hypothesis

)2/()()(

,i.e.

)2/(/

if level cesignificanat

offavor in Reject

210

120

2

10

10

n

n

txsxnt

tns

xt

HH

Selection of Often chosen as 0.05, 0.01, or 0.1Actually, Fisher said in 1956:– No scientific worker has a fixed level of

significance at which year to year, and in all circumstances, he rejects hypotheses; he rather gives his mind to each particular case in the light of his evidence and hid ideas

1313

1414

Evaluating Normality of Univariate Evaluating Normality of Univariate Marginal DistributionsMarginal Distributions

0026.0]3ˆ[

),( is ˆ ofon distributi The

),( large, is When

on distributi binomial

:intervalan within samples ofNumber

n

pqppP

n

pqpN

n

yp

npqnpNqpy

nn

qpy

n

yny

yny

1414

Confidence Interval for 0

nstXnstXCI

nstXnstX

nstXnstX

nstXnst

tns

X

nn

nn

nn

nn

n

/)025.0(,/)025.0(:

95.0

/)025.0(/)025.0(Pr

95.0

/)025.0(/)025.0(Pr

95.0

/)025.0(/)025.0(Pr

95.0)025.0(/

Pr

1195

11

11

11

10

1515

Neyman’s Interpretation

01616

Statistical Significance vs. Practical Significance

The cram school claims that its course will increase the IQ of your child statistically significant at the 0.05 levelAssume that 100 students took the courses were tested, and the population standard deviation is 15The actual IQ improvement to be statistically significant at 0.05 level is simply 94.296.15.1)/( 025.0 zn 1717

More Specific HypothesesNull hypothesis

Alternative hypothesis

00 : H

11 : H

1818

Type I and Type II Errors

1919

Type I and Type II ErrorsTruth

H0

(non-effective)

H1

(effective)

StudyResults

H0

(non-effective)

1- (type II error)

H1

(effective)

(type I error, false alarm)

1-(power)

2020

Power

The probability of concluding that the sample came from the H1 distribution (i.e., concluding there is a significant difference), when it really did from the H1 distribution (there is a difference)

2121

Power vs. Difference of Means

power

0

1

2222

n/01

Effective SizesHow many samples are required to validate the following claim of the cram school:– Our course will raise IQ levels of your child by 5

points

statistically significant at 0.05 level, and the type II error is 0.1Normal IQ mean is 100, with standard deviation 15Sample standard deviation is assumed to be 15, too

2323

Effective Sizes

95)(

/

28.1/

105

96.1/

100

as valuecriticalSet

2

zzn

zzn

zn

CV

zn

CV

CV

2424