hlth 300 biostatistics for public health practice, raul cruz-cano, ph.d

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HLTH 300 Biostatistics for Public Health Practice,

Raul Cruz-Cano, Ph.D.4/7/2014, Spring 2014

Fox/Levin/Forde, Elementary Statistics in Social Research, 12e

Chapter 7: Testing Differences between Means

1

2

Midterm Exam Results

Midterm ExamMorning Group Mean: 80.62Afternoon Group Mean: 85.65

HW#7: Add explanation, formula and example about the Coefficient of Variation and I’ll to multiply your grade by =1.05263


Distinguish between the null and research hypotheses

Understand the sampling distribution of differences between mean and use it to test hypotheses

Understand the rationale behind levels of significance

Test the differences between means

Understand the logic of one-tailed tests

CHAPTER OBJECTIVES

7.1

7.2

7.3

7.4

7.5

7.6 Calculate Cohen’s d

7.7 List the requirements for testing the differences between means

Distinguish between the null and research hypotheses

Learning ObjectivesAfter this lecture, you should be able to complete the following Learning Outcomes

7.1

5

7.1

We’ve learned that a population mean or proportion can be estimated

Researchers really want to test hypotheses•These hypotheses typically refer to differences between groups

In this chapter, we’ll learn how to test hypotheses about differences between sample means and proportions

Introduction

The Null and Research Hypotheses7.1

The Null Hypothesis

The Research

Hypothesisvs.

• There is no statistically significant difference between the sample means of two groups

• Any observed difference is the result of sampling error alone

• There is a statistically significant difference between the sample means of two groups

• A true population difference does exist

1 2 1 2

Understand the sampling distribution of differences between means and use it to test hypotheses


7.2

7.2

Figure 7.2

9

7.2

The sampling distribution approximates a normal curve

• This provides the basis for testing hypotheses between sample means

• We need to use standard scores or z scores

Testing Hypotheses with the Distribution of Differences between Means

21

21

XX

XXz

1 2

1

2

mean of the first sample

mean of the second sample standard deviation of the sampling distribution

of differences between meansX X

X

X

Understand the rationale behind levels of significance


7.3

7.3

Figure 7.7

12

7.3

Used to determine statistical significance Symbolized by αlpha• The level of probability where decisions can be made with

confidence

Levels of Significance

7.3

Figure 7.6

Test the differences between means


7.4

Tests of Difference

Proportions:Box 7.4, page 250

Means

21

2211*

21

21**

21

)1(21

21

NNPNPNP

NNNNPPs

sPPz

PP

PP

Dependent Samples

Same & Matched SamplesBox 7.2 &7.3, page 244&247

each) cases (two sample in the #1

1

)(

21

221

2

NNdfsXXt

Nss

XXND

s

D

DD

D

Independent SamplesKnown σ1 and σ2

Not realistic

Unknown σ1 and σ2

“Same” VarianceBox 7.1, page 239

Unequal Variancepage 243

2

2

,

2 and 2

21

21

21

21

21

222

211

22

2

222

22

11

212

1

1

2

2

1

21

21

NNdf

sXXt

NNNN

NNsNsNs

XNX

sXNX

s

ss

ss

XX

XX

),min(

11

,

2or 2

21

21

2

22

1

21

22

2

222

22

11

212

1

1

2

2

1

21

21

NNdf

sXXt

Ns

Nss

XNX

sXNX

s

ss

ss

XX

XX

Tests of Difference


Means

21

2211*

21

21**

21

)1(21

21

NNPNPNP

NNNNPPs

sPPz

PP

PP

Dependent Samples



Not realistic

Unknown σ1 and σ2



2

2

,

2 and 2

21

21

21

21

21

222

211

22

2

222

22

11

212

1

1

2

2

1

21

21

NNdf

sXXt

NNNN

NNsNsNs

XNX

sXNX

s

ss

ss

XX

XX

),min(

11

,

2or 2

21

21

2

22

1

21

22

2

222

22

11

212

1

1

2

2

1

21

21

NNdf

sXXt

Ns

Nss

XNX

sXNX

s

ss

ss

XX

XX

each) cases (two sample in the #

1

1

)(

21

221

2

NNdfsXXt

Nss

XXND

s

D

DD

D

17

7.4

Standard Error of the Differences between Means

• The standard deviation of the distributions of differences can be estimated

Testing differences between means• t is used instead of z because we don’t know the true

population standard deviation

Test the Differences between Means

1 2

2 21 1 2 2 1 2

1 2 1 22X X

N s N s N NsN N N N

1 2

1 2

X X

X Xts

18

Box 7.1 in page 239Problem 22

Tests of Difference


Means

21

2211*

21

21**

21

)1(21

21

NNPNPNP

NNNNPPs

sPPz

PP

PP

Dependent Samples



Not realistic

Unknown σ1 and σ2



2

2

,

2 and 2

21

21

21

21

21

222

211

22

2

222

22

11

212

1

1

2

2

1

21

21

NNdf

sXXt

NNNN

NNsNsNs

XNX

sXNX

s

ss

ss

XX

XX

),min(

11

,

2or 2

21

21

2

22

1

21

22

2

222

22

11

212

1

1

2

2

1

21

21

NNdf

sXXt

Ns

Nss

XNX

sXNX

s

ss

ss

XX

XX


1

1

)(

21

221

2

NNdfsXXt

Nss

XXND

s

D

DD

D

20

7.4

The formula for estimating the standard error of the differences between means pools variance information from both samples

• This assumes that the population variances are the same for the two groups

• If either sample variance is more than twice as large as the other, we should use the following formula that does not pool the variances

Adjustment for Unequal Variances

1 2

2 21 2

1 21 1X X

s ssN N

21

Example in page 243Problem 24

Tests of Difference


Means

21

2211*

21

21**

21

)1(21

21

NNPNPNP

NNNNPPs

sPPz

PP

PP

Dependent Samples



Not realistic

Unknown σ1 and σ2



2

2

,

2 and 2

21

21

21

21

21

222

211

22

2

222

22

11

212

1

1

2

2

1

21

21

NNdf

sXXt

NNNN

NNsNsNs

XNX

sXNX

s

ss

ss

XX

XX

),min(

11

,

2or 2

21

21

2

22

1

21

22

2

222

22

11

212

1

1

2

2

1

21

21

NNdf

sXXt

Ns

Nss

XNX

sXNX

s

ss

ss

XX

XX


1

1

)(

21

221

2

NNdfsXXt

Nss

XXND

s

D

DD

D

23

7.4

The before-after or panel design consists of a single sample measured at two points in time

• This means that the samples are no longer independent and therefore different formulas are required:

Comparing Dependent Samples

1 2

D

X Xts

1D

D

ssN

2

2

1 2D

Ds X X

N

standard deviation of the distribution of before-after differences scores

after raw score subtracted from before raw score number of cases or respondents in sample

DS

DN

24

Box 7.2 in page 244 and Box 7.3 in page 247Problem 34

Tests of Difference


Means

21

2211*

21

21**

21

)1(21

21

NNPNPNP

NNNNPPs

sPPz

PP

PP

Dependent Samples



Not realistic

Unknown σ1 and σ2



2

2

,

2 and 2

21

21

21

21

21

222

211

22

2

222

22

11

212

1

1

2

2

1

21

21

NNdf

sXXt

NNNN

NNsNsNs

XNX

sXNX

s

ss

ss

XX

XX

),min(

11

,

2or 2

21

21

2

22

1

21

22

2

222

22

11

212

1

1

2

2

1

21

21

NNdf

sXXt

Ns

Nss

XNX

sXNX

s

ss

ss

XX

XX


1

1

)(

21

221

2

NNdfsXXt

Nss

XXND

s

D

DD

D

26

7.4

The logic for testing the differences between two proportions is the same as when dealing with means

• The formulas are just different

Two Sample Test of Proportions

1 2

1 2

P P

P Pzs

1 1 2 2

1 2

*N P N PPN N

1 2

1 2

1 2

* 1 *P PN Ns P PN N

27

Box 7.4 in page 250Problem 41

Understand the logic of one-tailed tests


7.5

One-Tailed Tests7.5

Two-Tailed One-Tailedvs.

1 2

1 2

Null Hypothesis:

Research Hypothesis:

1 2

1 2

Null Hypothesis:

Research Hypothesis:

7.5

Figure 7.9

31

Let’s review the examples, this time looking at the one-tailed test hypothesisNotice that now we use page 553Box 7.5 in page 254 Same sampleBox 7.6 in page 256 Independent Samples

List the requirements for testing the difference between means


7.7

Requirements for Testing the Differences between Means7.7

A Comparison between Two Means

Interval Data

Random Sampling

A Normal Distribution

Equal Variances

34

Homework

Chapter 7 Problems: 27, 28, 36


When testing for differences between means (or proportions), researchers begin with a null and

research hypothesis

The logic of the distribution of differences between means is integral for hypothesis testing

The level of significance determines the level of probability at which the null hypothesis can be

rejected with confidence

CHAPTER SUMMARY

7.1

7.2

7.3


Researchers can use several different methods to test for differences between means and proportions

CHAPTER SUMMARY

7.4

One-tailed tests are used when the direction of a relationship is anticipated

Cohen’s d can be calculated to determine effect size

7.5

7.6

There are several requirements that must be met in order to test the differences between means7.7

hlth 300 biostatistics for public health practice, raul cruz-cano, ph.d

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