1 Basic Statistics

Introduction

Estimation - Condence Interval

Hypothesis testing - p value

Determination of sample size

Summary

Basic statistics: p value and condence interval

Nguyen Quang Vinh

February, 2012

Nguyen Quang Vinh Basic statistics: p value and condence interval

Introduction




Summary

Outline

1

Introduction

Statistics

Objectives Statistics2


Estimation

A point estimate

An interval estimate

Interpretation a condence interval

3


Hypothesis testing

Hypotheses

p value

The hypothesis testing procedure

Power of Test


Introduction




Summary

Statistics

Objectives Statistics

Outline

1

Introduction

Statistics



Estimation

A point estimate



3


Hypothesis testing

Hypotheses

p value


Power of Test


Introduction




Summary

Statistics


Statistics

Statistics:

science of data

study of uncertainty

Biostatistics: data from: Medicine, Biological sciences

(business, education, psychology, agriculture, economics...)

Modern society:

Reading

Writing

Statistical thinking: to make the strongest possible conclusions

from limited amounts of data.


Introduction




Summary

Statistics


Outline

1

Introduction

Statistics



Estimation

A point estimate



3


Hypothesis testing

Hypotheses

p value


Power of Test


Introduction




Summary

Statistics



Objectives:

(1) Organize & summarize data

(2) Reach inferences: sample population

Statistics:Descriptive statistics(1)Inferential statistics: drawing of inferences(2)Estimation (point estimate & interval estimate condenceinterval)

Hypothesis testing reaching a decision (p value)Parametric statistics

Non-parametric statistics

Introduction




Summary

Estimation

A point estimate



Outline

1

Introduction

Statistics



Estimation

A point estimate



3


Hypothesis testing

Hypotheses

p value


Power of Test


Introduction




Summary

Estimation

A point estimate



Why estimation?

Two reasons:

Innite populations: incapable of complete examination

Finite populations: cost, time

In addition, estimation can help not to defer a conclusion, until

the entire population has been observed


Introduction




Summary

Estimation

A point estimate



Estimation of

mean(s):

a single population mean

the dierence between two population means: unpaired, paired

proportion(s):

a single population proportion

the dierence of two population proportions

variance(s):

a single population variance

the ratio of two population variances


Introduction




Summary

Estimation

A point estimate



An estimation of these parameters

An estimation of these parameters:

Point estimate

Interval estimate


Introduction




Summary

Estimation

A point estimate



Outline

1

Introduction

Statistics



Estimation

A point estimate



3


Hypothesis testing

Hypotheses

p value


Power of Test


Introduction




Summary

Estimation

A point estimate



A point estimate

Estimator Parameter

In many cases, a parameter may be estimated by more than one

estimator.

Example:

Sample mean estimate population meanSample median estimate population mean


Introduction




Summary

Estimation

A point estimate



The criteria of good estimator (opt.)

(1) E (x) = without systematic errorE (x) is called systematic error(2) Mean square error

E (x)2 must be small in comparison to

E(x)2

must be small


Introduction




Summary

Estimation

A point estimate



Outline

1

Introduction

Statistics



Estimation

A point estimate



3


Hypothesis testing

Hypotheses

p value


Power of Test


Introduction




Summary

Estimation

A point estimate




In general, an interval estimate is obtained by the formula:

estimator (reliability coecient) x (standard error)

What is dierent is the source of the reliability coecient:

In particular, when sampling is from a normal distribution with

known variance, an interval estimate for may be expressedas: x z/2x


Introduction




Summary

Estimation

A point estimate



Outline

1

Introduction

Statistics



Estimation

A point estimate



3


Hypothesis testing

Hypotheses

p value


Power of Test


Introduction




Summary

Estimation

A point estimate



How to interpret the interval given by this expression

In repeated sampling 100(1)% of all intervals of the formwill in the long run include the population mean, The quantity (1), is called the condence coecient &The interval x z/2x , is called the condence interval for The most frequently used values are: .90, .95, .99, which have

associated reliability factors, respectively, of 1.645, 1.96, 2.58


Introduction




Summary

Estimation

A point estimate



The practical interpretation

We are 100(1)% condent that the single computedinterval x z/2x contains the population mean, Example: ...

E = margin error = maximum error = practical / clinical

acceptable error:

E = z/2x = z/2 n


Introduction




Summary

Hypothesis testing

Hypotheses

p value


Power of Test

Outline

1

Introduction

Statistics



Estimation

A point estimate



3


Hypothesis testing

Hypotheses

p value


Power of Test


Introduction




Summary

Hypothesis testing

Hypotheses

p value


Power of Test

Why hypothesis testing

Hypothesis (H.): a statement concerns about some one or

more populations

Testing hypothesis: to aid researcher in reaching a decision

concerning a population by examining a sample from that

population


Introduction




Summary

Hypothesis testing

Hypotheses

p value


Power of Test

Hypothesis testing for

mean(s)

a single population mean

the dierence between two population means: unpaired, paired

proportion(s)

a single population proportion

unpaired: a small sample, a suciently large sample

paired

the dierence of two population proportions

variance(s)

a single population variance

the dierence of two population variances


Introduction




Summary

Hypothesis testing

Hypotheses

p value


Power of Test

Outline

1

Introduction

Statistics



Estimation

A point estimate



3


Hypothesis testing

Hypotheses

p value


Power of Test


Introduction




Summary

Hypothesis testing

Hypotheses

p value


Power of Test

Two types of hypotheses

(1) Research Hypotheses:

The conjecture or supposition

It may be the results of years of observation

Research H. leads directly to Statistical H.

(2) Statistical Hypotheses: Hypotheses are stated in such a way

that they may be evaluated by appropriate statistical techniques.

H

O

H

A


Introduction




Summary

Hypothesis testing

Hypotheses

p value


Power of Test

Statistical Hypotheses

H

O

The H

O

is the hypothesis that is tested

The H

O

should contain either =,,(The statement concerns about some one or more population's

parameters in term of equality or inequality)

H

A

What we hope or expect to be able to conclude as a result of

the test usually should be placed in the H

A

The H

O

& HA

are complementary

One-sided vs. Two-sided Hypothesis Tests (opt.)


Introduction




Summary

Hypothesis testing

Hypotheses

p value


Power of Test

Notes

Neither hypothesis testing nor statistical inference, in general,

leads to proof a hypothesis

It merely indicates whether the hypothesis is supported or not

supported by the available data

When we fail to reject the H

O

, we do not say that it is true,

but that it may be true

When we speak of accepting a H

O

, we have this limitation

in mind & do not wish to convey the idea that accepting

implies proof


Introduction




Summary

Hypothesis testing

Hypotheses

p value


Power of Test

Outline

1

Introduction

Statistics



Estimation

A point estimate



3


Hypothesis testing

Hypotheses

p value


Power of Test


Introduction




Summary

Hypothesis testing

Hypotheses

p value


Power of Test

p value

Test statistic =p value

General formula:

Teststatistic = relevantstatistichypothesizedparameterS .E .oftherelevantstatistic

Example: z = x0n

Test statistic p valueDecision maker, since the decision to reject or not to reject the

H

O

depends on the magnitude of the test statistic


Introduction




Summary

Hypothesis testing

Hypotheses

p value


Power of Test

Decision rule for a rejection or not the H

O

= type I error = level of signicance (say, .01, .05, .10) = type II error (say, .05, .10, .20)When we reject a H

O

p < , risk of committing a type Ierror, rejecting a true H

O

When we fail to reject a H

O

: risk of committing a type II

error, accepting a false H

O


Introduction




Summary

Hypothesis testing

Hypotheses

p value


Power of Test

Type I & Type II error

Conditions under which type I & type II errors may be committed (the four possibilities)

Actual Situation(Truth in the population)

Ho false Ho true

The results in the study sample Conclusion:

RejectHo

Correct decision

Type I error

Fail toreject Ho

Type II error

Correct decision


Introduction




Summary

Hypothesis testing

Hypotheses

p value


Power of Test

Testing Hypothesis Rejected or not rejected HO

In the testing process the H

O

either is rejected or is not

rejected

If H

O

is not rejected, we will say that the data on which the

test is based do not provide sucient evidence to cause

rejection

If the testing process leads to rejection, we will say that the

data at hand are not compatible with the H

O

, but are

supportive of some other hypothesis & may be designated by

H

A

(H

A

a contradiction statement of H

O

)


Introduction




Summary

Hypothesis testing

Hypotheses

p value


Power of Test

Outline

1

Introduction

Statistics



Estimation

A point estimate



3


Hypothesis testing

Hypotheses

p value


Power of Test


Introduction




Summary

Hypothesis testing

Hypotheses

p value


Power of Test

The Five-Step practical procedure for Hypothesis Testing

(opt.)

Step 1: Set up H

O

, H

A

1. Data: The nature of the data (classication)

2. Assumptions: The normality of the population distribution,

equality of variances, independence of samples. . .

3. Hypotheses: H

O

, H

A

Step 2: Dene the test statistic

4. Test statistic

5. Distribution of the Test Statistic


Introduction




Summary

Hypothesis testing

Hypotheses

p value


Power of Test

The Five-Step practical procedure for Hypothesis Testing,

cont. (opt.)

Step 3: Dene a rejection region: having determined a value

for 6. Decision rule

Step 4:

7. Calculate the value of the test statistic, and compare it with

the acceptance & rejection regions that have already been

specied.

8. State our decision: to reject H

O

or to fail to reject H

O

Step 5:

9. Give a conclusion: this statement should be free of

statistical jargon & should merely summarize the results of the

analysis.


Introduction




Summary

Hypothesis testing

Hypotheses

p value


Power of Test

Outline

1

Introduction

Statistics



Estimation

A point estimate



3


Hypothesis testing

Hypotheses

p value


Power of Test


Introduction




Summary

Hypothesis testing

Hypotheses

p value


Power of Test

The Power of a Statistical Test (opt.)

The probability of a type II error, b, has remained a phantom:

we know it is there,

but we don't know what it is

One thing we can say is that: a wide C.I. for m means that the

corresponding 2-tailed test of Ho versus HA has a large chance

of failing to reject a false Ho; that is b is large.


Introduction




Summary

Hypothesis testing

Hypotheses

p value


Power of Test

Determining b (opt.)

b = P(fail to reject H

O

when H

O

is false)

1 - b = P(rejecting HO

when H

O

is false)

1 - b represents the probability of making a correct decision in

the event that H

O

is false

Since we like b to be small, that is we prefer 1 - b to be large

The value of 1 - b is referred to as the power of test


Introduction




Summary

Hypothesis testing

Hypotheses

p value


Power of Test

Power of test (opt.)

Power of test = P(Z>z1) + P(Zz1)P(Z

Introduction




Summary

Determination of sample size (opt.)

Estimating a condence interval

Testing a hypothesis


Introduction




Summary

Summary

1. Statistics:

Descriptive statisticsorganize & summarize dataInferential statistics drawing of inferencesEstimation

Hypothesis testing

Modeling

2. Estimation - condence interval, estimator

3. Hypothesis testing - p value


IntroductionStatisticsObjectives Statistics

Estimation - Confidence IntervalEstimationA point estimateAn interval estimateInterpretation a confidence interval

Hypothesis testing - p valueHypothesis testingHypothesesp valueThe hypothesis testing procedurePower of Test

1 Basic Statistics

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