Chapter 13 – 1

Chapter 9: Testing Hypotheses

• Overview• Research and null hypotheses• One and two-tailed tests• Errors• Testing the difference between two means• t tests

Chapter 13 – 2

Overview

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Independent Variables

Nominal Interval

Considers the distribution of one variable across the categories of another variable

Considers the difference between the mean of one group on a variable with another group

Considers how a change in a variable affects a discrete outcome

Considers the degree to which a change in one variable results in a change in another

You already know how to deal with two

nominal variables

Chapter 13 – 3

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Independent Variables

Nominal Interval

Considers the difference between the mean of one group on a variable with another group

Considers how a change in a variable affects a discrete outcome

Considers the degree to which a change in one variable results in a change in another

You already know how to deal with two

nominal variables

Lambda

TODAY! Testing the differences

between groups

Overview

Chapter 13 – 4

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Independent Variables

Nominal Interval

Considers how a change in a variable affects a discrete outcome

Considers the degree to which a change in one variable results in a change in another

You already know how to deal with two

nominal variables

Lambda

TODAY! Testing the differences

between groups

Confidence Intervalst-test

Overview

Chapter 13 – 5

Example

• Draw a random sample of 100 Africa American from GSS 1998.

• Calculate the mean earnings--$24,100• Based on census information, the mean earnings

for Americans is $28,985.• Is the observed gap ($28,985 - $24,100) large

enough to convince us that the sample we drew is not representative of the population?

Chapter 13 – 6

Example

• The average earnings of the Africa Americans are indeed lower than the national average

• The average earnings of the Africa Americans are about the same as the national average, and this sample happens to show a particularly low mean.

Chapter 13 – 7

General ExamplesIn

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Independent Variables

Nominal Interval

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Independent Variables

Nominal Interval

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Independent Variables

Nominal Interval

Is one group scoring significantly higher on average than another group?

Is a group statistically different from another on a particular dimension?

Is Group A’s mean higher than Group B’s?

Chapter 13 – 8

Specific ExamplesIn

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Independent Variables

Nominal Interval

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Independent Variables

Nominal Interval

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Independent Variables

Nominal Interval

Do people living in rural communities live longer than those in urban or suburban areas?

Do students from private high schools perform better in college than those from public high schools?

Is the average number of years with an employer lower or higher for large firms (over 100 employees) compared to those with fewer than 100 employees?

Chapter 13 – 9

• Statistical hypothesis testing – A procedure that allows us to evaluate hypotheses about population parameters based on sample statistics.

• Research hypothesis (H1) – A statement reflecting the substantive hypothesis. It is always expressed in terms of population parameters, but its specific form varies from test to test.

• Null hypothesis (H0) – A statement of “no difference,” which contradicts the research hypothesis and is always expressed in terms of population parameters.

Testing Hypotheses

Chapter 13 – 10

Research and Null HypothesesOne Tail — specifies the hypothesized direction• Research Hypothesis:

H1: 2 1, or 2 1 > 0 • Null Hypothesis:

H0: 2 1, or 2 1 = 0

Two Tail — direction is not specified (more common)• Research Hypothesis:

H1: 2 1, or 2 1 = 0• Null Hypothesis:

H0: 2 1, or 2 1 = 0

Chapter 13 – 11

One-Tailed Tests• One-tailed hypothesis test – A hypothesis test

in which the alternative is stated in such a way that the probability of making a Type I error is entirely in one tail of a sampling distribution.

• Right-tailed test – A one-tailed test in which the sample outcome is hypothesized to be at the right tail of the sampling distribution.

• Left-tailed test – A one-tailed test in which the sample outcome is hypothesized to be at the left tail of the sampling distribution.

Chapter 13 – 12

Two-Tailed Tests

• Two-tailed hypothesis test – A hypothesis test in which the region of rejection falls equally within both tails of the sampling distribution.

Chapter 13 – 13

Probability Values

• Z statistic (obtained) – The test statistic computed by converting a sample statistic (such as the mean) to a Z score. The formula for obtaining Z varies from test to test.

• P value – The probability associated with the obtained value of Z.

Chapter 13 – 14

Probability Values

Chapter 13 – 15

Probability Values

• Alpha ( ) – The level of probability at which the null hypothesis is rejected. It is customary to set alpha at the .05, .01, or .001 level.

Chapter 13 – 16

Five Steps to Hypothesis Testing

(1) Making assumptions

(2) Stating the research and null hypotheses and selecting alpha

(3) Selecting the sampling distribution and specifying the test statistic

(4) Computing the test statistic

(5) Making a decision and interpreting the results

Chapter 13 – 17

• Type I error (false rejection error)the probability (equal to ) associated with rejecting a true null hypothesis.

• Type II error (false acceptance error)the probability associated with failing to reject a false null hypothesis.

Based on sample results, the decision made is to…reject H0 do not reject H0

In the true Type I correct population error () decisionH0 is ...

false correct Type II error decision

Type I and Type II Errors

Chapter 13 – 18

One-Sample z Test• When we know population parameters μ

and σ, how likely we could draw a random sample whose mean (y bar) differs from μ?

• Null Hypothesis

Population mean μy equals to population mean μ.

Chapter 13 – 19

One-Sample z Test

• Test statistic

0

1

: =

: Y

Y

H

H

Y-z=

/ n

Chapter 13 – 20

One-Sample z Test• Compare z we calculate to the critical value

• Make a decisioncritical| | v.s. zz

critical

0

critical

0

| | z

Reject H

| | z

Not reject H

z

z

Chapter 13 – 21

3.55,

Y=3.41

.21

Y-z=

/ n

•Example

how likely we could draw a random sample from a population whose mean is differ from μ? id GPA

7 3.6

1 3.2

1 3.2

1 3.2

4 3.4

5 3.5

7 3.6

6 3.5

7 3.6

3 3.3

Chapter 13 – 22

Example

• Is the observed gap ($28,985 - $24,100) large enough to convince us that the sample we drew is not representative of the population?

Chapter 13 – 23

Five-step Testing Hypothesis-1

Making Assumptions:

• A random sample is selected.

• Because N>50, the assumption of normal population is not required.

• The level of measurement of the dependent variable is interval-ratio.

Chapter 13 – 24

Five-step Testing Hypothesis-2Stating the Research and the Null Hypotheses• The research hypothesis is

• The null hypothesis is

1 Y: $28,985H

0 Y: $28,985H

Chapter 13 – 25

Five-step Testing Hypothesis-3

Selecting the Sampling distribution and Specify the Test Statistic

• We use the z distribution and the z statistic to test the null hypothesis

Chapter 13 – 26

Five-step Testing Hypothesis-4

Computing the z Test Statistic

critical

$24,100 N=100

$28,985 23,335

24,100 28,9852.09

/ 23,335 / 100

| | =2.09 > z 1.96 if =.05

.0183

Y

Yz

N

z

P

Chapter 13 – 27

Five-step Testing Hypothesis-5

Making a Decision and Interpreting the Results• Our obtained |z| statistic of 2.09 is greater than

1.96 or probability of obtaining a z statistic of 2.09 is less than .05. This P value is below .05 alpha level.

• The probability of obtaining the difference of $4885 ($28,985 - $24,100) between the income of African Americans and the national average for all, if the null hypothesis were true, is extremely low.

Chapter 13 – 28

Five-step Testing Hypothesis-5

• We have sufficient evidence to reject the null hypothesis and conclude that the average earnings of African American are significantly different from the average earnings of all. The difference is significant at the .05 level.

Chapter 13 – 29

• t statistic (obtained) – The test statistic computed to test the null hypothesis about a population mean when the population standard deviation is unknown and is estimated using the sample standard deviation.

• t distribution – A family of curves, each determined by its degrees of freedom (df). It is used when the population standard deviation is unknown and the standard error is estimated from the sample standard deviation.

• Degrees of freedom (df) – The number of scores that are free to vary in calculating a statistic.

t Test

Chapter 13 – 30

One-Sample t Test• t test

A test of significance similar to the z test but used when the population’s standard deviation is unknown.

2

Y-t= , df=n-1

/

( )where

1

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Y YS

n

Chapter 13 – 31

t distribution

Chapter 13 – 32

t distribution table

Chapter 13 – 33

Chapter 13 – 34

3.55,

Y=3.41

df=9

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/YS n

•Example

how likely we could draw a random sample whose mean (Y bar) differs from μ?

id GPA

7 3.6

1 3.2

1 3.2

1 3.2

4 3.4

5 3.5

7 3.6

6 3.5

7 3.6

3 3.3

Chapter 13 – 35

The Earnings of White Women

• We drew a sample of white females (N=371) from GSS 2002.

• The mean earnings is $28,889 with a standard deviation 21,071.

• In 2002, the national average earnings for all women is $24,146.

Chapter 13 – 36

Five-step Testing Hypothesis-1

Making Assumptions:

• A random sample is selected.

• The sample size is large.

• The level of measurement of the dependent variable is interval-ratio.

Chapter 13 – 37

Five-step Testing Hypothesis-2Stating the Research and the Null Hypotheses• The research hypothesis is

• The null hypothesis is

1 Y: $24,146H

0 Y: $24,146H

Chapter 13 – 38

Five-step Testing Hypothesis-3

Selecting the Sampling distribution and Specify the Test Statistic

• We use the t distribution and the t statistic to test the null hypothesis

Chapter 13 – 39

Five-step Testing Hypothesis-4

Computing the Test Statistic• Firstly, calculate the degree of freedom associated

with test

33.4371/21071

2414628889

/

37013711

NS

Yt

Ndf

Y

Chapter 13 – 40

Five-step Testing Hypothesis-5

Making a Decision and Interpreting the Results• Our obtained t statistic of 4.33 is greater than

1.980 or probability of obtaining a t statistic of 4.33 is less than .05. This P value is below .05 alpha level.

• The probability of obtaining the difference of $4743 ($28889-$24146) between the income of white women and the national average for all women, if the null hypothesis were true, is extremely low.

Chapter 13 – 41

Five-step Testing Hypothesis-5

• We have sufficient evidence to reject the null hypothesis and conclude that the average earnings of white women are significantly different from the average earnings of all women. The difference is significant at the .05 level.

Chapter 13 – 42

Exercise

• Can you do a one-tail test see if the mean earnings of white women is significantly higher than the average for all women?

Chapter 13 – 43

Two-Sample t Tests

• The t-test assesses whether the means of two populations statistically differ from each other.

• The 2 independent sample t-test is used when testing 2 independent groups..

Chapter 13 – 44

t-test for difference between two meansIs the value of 2 1 significantly different from 0?This test gives you the answer:

If the t value is greater than 1.96, the difference between the means is significantly different from zero at an alpha of .05 (or a 95% confidence level).

The difference between the two means

the estimated standard error of the difference

21

21

21

)2(

YY

NN S

YYt

The critical value of t will be higher than 1.96 if the total N is less than 122. See Appendix C for exact critical values when N < 122.

Chapter 13 – 45

Test Statistic• Equal population variance assumed

2

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Chapter 13 – 46

Test Statistic• Unequal population variance assumed

112

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Chapter 13 – 47

t-test and Confidence Intervals

21

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YY

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The t-test is essentially creating a confidence interval around the difference score. Rearranging the above formula, we can calculate the confidence interval around the difference between two means:

)(2121 YY

StYY If this confidence interval overlaps with zero, then we cannot be certain that there is a difference between the means for the two samples.

Chapter 13 – 48

Why a t score and not a Z score?

• Use of the Z distribution has assumes the population standard error of the difference is known. In practice, we have to estimate it and so we use a t score.

• When N gets larger than 50, the t distribution converges with a Z distribution so the results would be identical regardless of whether you used a t or Z.

• In most sociological studies, you will not need to worry about the distinction between Z and t.

)(2121 YY

StYY

Chapter 13 – 49

t-Test Example 1Mean pay according to gender:

N Mean Pay S.D.

Women 46 $10.29 .8766

Men 54 $10.06 .9051

9824654df

.8609.2671

.23

46

1

54

1

24654

.8766146.9051154

10.2910.06T

22

What can we conclude about the difference in

wages?

Equal population variances assumed

Chapter 13 – 50

Mean pay according to gender:

N Mean Pay S.D.

Women 57 $9.68 1.055

Men 51 $10.32 .9461

06127551df

7484.0.8552

64.

57

1

51

1

27551

055.11759461.151

68.932.1022

T

What can we conclude about the difference in

wages?

t-Test Example 2

Equal population variances assumed

Chapter 13 – 51

Using these GSS income data, calculate a t-test statistic to determine if the difference between the two group means is statistically significant.

In-Class Exercise

Mean Standard Deviation N

Men $22,052.51 $17,734.92 434

Women $14,331.21 $12,165.89 448

62.71,027.18

7,831.30

44889.165,12

43492.734,17

21.221,1451.052,22T

220)88(

2

22

1

21

12

2)N(N

Ns

Ns

YYT

21

Unequal population variances assumed

Chapter 13 – 52

Steps(1) Making assumptions

(2) Stating the research and null hypotheses and selecting alpha

(3) Selecting the sampling distribution and specifying the test statistic

(4) Computing the test statistic

(5) Making a decision and interpreting the results

Chapter 13 – 53

Example

• Suppose we have obtained # of years of education from one random sample of 38 police officers from City A and # of years of education from a second random sample of 30 police officers from City B.

• The average years of education for the sample from City A is 15 with a standard deviation of 2.

• The average years of education for the sample from City B is 14 with a standard deviation of 2.5.

• Is there a statistically significant difference between the education levels of police officers in City A and City B?

Chapter 13 – 54

1.Making Assumptions

• Two random samples are selected.

• The sample sizes are large. Because N>50, the assumption of normal population is not required.

• The level of measurement of the dependent variable is interval-ratio.

• Population variances are assumed to be equal.

Chapter 13 – 55

2.State Hypotheses

H0: There is no statistically significant difference between the mean education level of police officers working in City A and the mean education level of police officers working in City B.

Chapter 13 – 56

2.State Hypotheses

For a 2-tailed hypothesis test

• H1: There is a statistically significant difference between the mean education level of police officers working in City A and the mean education level of police officers working in City B.

Chapter 13 – 57

2.State Hypotheses

For a 1-tailed hypothesis test

• H1: The mean education level of police officers working in City A is significantly greater than the mean education level of police officers working in City B.

Chapter 13 – 58

2. Set the Rejection Criteria

• Determine the degrees of freedom df = (n1+n2)-2 df = 38+30-2=66• Determine level of confidence -- alpha (1 or 2-

tailed test) Use the t-distribution table to determine the critical

value If using 2-tailed test Alpha.05, tcv= 1.997 If using 1-tailed test Alpha.05, tcv= 1.668

Chapter 13 – 59

3. Specifying the test statistic

• Because the population variances are unknown, t-distribution should be used.

• t-statistic.

Chapter 13 – 60

4. Compute Test Statistic

6623038

835.1545.

1

30

1

38

1

23038

25.61304138

1415

11

2

11

2121

2

22

2

11

21

df

nnnn

snsn

YYt

Chapter 13 – 61

4. Compare the t-cal with t-cri

criticalcalculated

critical

calculated

tt

t

t

997.1

testtailed- twoif

835.1

criticalcalculated

critical

calculated

tt

t

t

668.1

testtailed-one if

835.1

Chapter 13 – 62

5. Make a decision

If using 2-tailed test• the test statistic 1.835 does not meet or

exceed the critical value of 1.997 for a 2-tailed test.

• There is no statistically significant difference between the mean years of education for police officers in City A and mean years of education for police officers in City B.

Chapter 13 – 63

If using 1-tailed test

• the test statistic 1.835 does exceed the critical value of 1.668 for a 1-tailed test.

• Police officers in City A have significantly more years of education than police officers in City B.

Chapter 13 – 64

Another Example

• http://www.gallup.com/poll/111703/Final-Presidential-Estimate-Obama-55-McCain-44.aspx

Chapter 13 – 65

Test for two sample proportions

21

2211

21

21

ˆ

11ˆ1ˆ

nn

pnpn

nn

ppz

Chapter 13 – 66

Interpreting a t testGroup Statistics

38 15.00 2.027 .329

30 13.83 2.534 .463

CITYa

b

EDUN Mean Std. Deviation

Std. ErrorMean

Independent Samples Test

.807 .372 2.110 66 .039 1.167 .553 .063 2.270

2.056 54.751 .045 1.167 .568 .029 2.304

Equal variancesassumed

Equal variancesnot assumed

EDUF Sig.

Levene's Test forEquality of Variances

t df Sig. (2-tailed)Mean

DifferenceStd. ErrorDifference Lower Upper

95% ConfidenceInterval of the

Difference

t-test for Equality of Means