dr. ka-fu wong

Ka-fu Wong © 2003 Chap 11- 1

Dr. Ka-fu Wong

ECON1003Analysis of Economic Data


Control GroupExperimental Group

Sample1

Sample2

To test the effect of an herbal treatment on improvement of memory you randomly select two samples, one to receive the treatment and one to receive a placebo. Results of a memory test taken one month later are given.

95

15

77

1

1

1

n

s

x

105

12

73

2

2

2

n

s

x

The resulting test statistic is 77 - 73 = 4. Is this difference significant or is it due to chance (sampling error)?

Treatment Placebo

Overview

Ka-fu Wong © 2003 Chap 11- 3l

GOALS

1. Understand the difference between dependent and independent samples.

2. Conduct a test of hypothesis about the difference between two independent population means when both samples have 30 or more observations.

3. Conduct a test of hypothesis about the difference between two independent population means when at least one sample has less than 30 observations.

4. Conduct a test of hypothesis about the mean difference between paired or dependent observations.

5. Conduct a test of hypothesis regarding the difference in two population proportions.

Chapter ElevenTwo Sample Tests of Two Sample Tests of HypothesisHypothesis


Two Sample Tests

TEST FOR EQUAL VARIANCESTEST FOR EQUAL VARIANCES TEST FOR EQUAL MEANSTEST FOR EQUAL MEANS

HHo

HH1

Population 1

Population 2

Population 1

Population 2

HHo

HH1

Population 1

Population 2

Population 1Population 2


The formula of general test statistic

Suppose we are interested in testing the population parameter () is equal to k. H0: = k H1: k

First, we need to get a sample estimate (q) of the population parameter ().

Second, we know in most cases, the test statistics will be in the following form: t=(q-k)/q

The form of q depends on what q is. Sample size and the null at hand determine the

distribution of the statistic. If is population mean, and the sample size is

larger than 30, t is approximately normal.


Comparing two populations

We wish to know whether the distribution of the differences in sample means has a mean of 0.

If both samples contain at least 30 observations we use the z distribution as the test statistic.


Hypothesis Tests for Two Population Means

Format 1Format 1

Two-Tailed Two-Tailed TestTest

Upper Upper One-Tailed One-Tailed TestTest

Lower Lower One-Tailed One-Tailed TestTest

0.0:

0.0:

21

210

AH

H

0.0:

0.0:

21

210

AH

H

0.0:

0.0:

21

210

AH

H

Format 2Format 2

21

210

:

:

AH

H

21

210

:

:

AH

H

21

210

:

:

AH

H

Preferred


Two Independent Populations: Examples

1. An economist wishes to determine whether there is a difference in mean family income for households in two socioeconomic groups. Do HKU students come from families with

higher income than CUHK students?

2. An admissions officer of a small liberal arts college wants to compare the mean SAT scores of applicants educated in rural high schools & in urban high schools.

Do students from rural high schools have lower A-level exam score than from urban high schools?


Two Dependent Populations: Examples

1. An analyst for Educational Testing Service wants to compare the mean GMAT scores of students before & after taking a GMAT review course.

Get HKU graduates to take A-Level English and Chinese exam again. Do they get a higher A-Level English and Chinese exam score than at the time they enter HKU?

2. Nike wants to see if there is a difference in durability of 2 sole materials. One type is placed on one shoe, the other type on the other shoe of the same pair.


Thinking Challenge

1. Miles per gallon ratings of cars before & after mounting radial tires

2. The life expectancies of light bulbs made in two different factories

3. Difference in hardness between 2 metals: one contains an alloy, one doesn’t

4. Tread life of two different motorcycle tires: one on the front, the other on the back

Are they independent or dependent?

independent

independent

dependent

dependent


Comparing two populations

No assumptions about the shape of the populations are required.

The samples are from independent populations.Values in one sample have no influence

on the values in the other sample(s).Variance formula for independent

random variables A and B: V(A-B) = V(A) + V(B)

The formula for computing the value of z is:

2

22

1

21

21

ns

ns

XXz


EXAMPLE 1

Two cities, Bradford and Kane are separated only by the Conewango River. There is competition between the two cities. The local paper recently reported that the mean household income in Bradford is $38,000 with a standard deviation of $6,000 for a sample of 40 households. The same article reported the mean income in Kane is $35,000 with a standard deviation of $7,000 for a sample of 35 households. At the .01 significance level can we conclude the mean income in Bradford is more?


EXAMPLE 1 continued

Step 1: State the null and alternate hypotheses.

H0: µB ≤ µK ; H1: µB > µK

Step 2: State the level of significance. The .01 significance level is stated in the problem.

Step 3: Find the appropriate test statistic. Because both samples are more than 30, we can use z as the test statistic.


Example 1 continued

Step 4: State the decision rule. The null hypothesis is rejected if z is greater than 2.33.

33.2z0

Rejection Region = 0.01

H0: µB ≤ µK ;

H1: µB > µK

Probability density of z statistic : N(0,1)

Acceptance Region = 0.01


Example 1 continued

Step 5: Compute the value of z and make a decision.

98.1

35)000,7($

40)000,6($

000,35$000,38$22

z

33.2z0

H0: µB ≤ µK ;

H1: µB > µK

1.98




Example 1 continued

The decision is to not reject the null hypothesis. We cannot conclude that the mean household income in Bradford is larger.


Example 1 continued

The p-value is:P(z > 1.98) = .5000 - .4761

= .0239

33.2z0


H0: µB ≤ µK ;

H1: µB > µK

1.98

P-value = 0.0239


Small Sample Tests of Means

The t distribution is used as the test statistic if one or more of the samples have less than 30 observations.

The required assumptions are:1. Both populations must follow the

normal distribution.2. The populations must have equal

standard deviations.3. The samples are from independent

populations.


Small sample test of means continued

Finding the value of the test statistic requires two steps.Step 1: Pool the sample standard deviations.

2

)1()1(

21

222

2112

nn

snsnsp

21

2

21

11nn

s

XXt

p

Step 2: Determine the value of t from the following formula.


EXAMPLE 2

A recent EPA study compared the highway fuel economy of domestic and imported passenger cars. A sample of 15 domestic cars revealed a mean of 33.7 mpg with a standard deviation of 2.4 mpg. A sample of 12 imported cars revealed a mean of 35.7 mpg with a standard deviation of 3.9.

At the .05 significance level can the EPA conclude that the mpg is higher on the imported cars?


Example 2 continued


H0: µD ≥ µI ; H1: µD < µI


Step 3: Find the appropriate test statistic. Both samples are less than 30, so we use the t distribution.


EXAMPLE 2 continued

Step 4: The decision rule is to reject H0 if t<-1.708. There are 25 degrees of freedom.

708.1t 0


05.0

:

:0

IDA

ID

H

H

Probability density of t statistic : t (df=25)


EXAMPLE 2 continued

918.921215

)9.3)(112()4.2)(115(

2

))(1())(1(

22

21

222

2112

nn

snsnsp

Step 5: We compute the pooled variance:


Example 2 continued

We compute the value of t as follows.

640.1

121

151

312.8

7.357.33

11

21

2

21

nns

XXt

p


Example 2 continued

708.1t 0


05.0

:

:0

IDA

ID

H

H

-1.640

H0 is not rejected. There is insufficient sample evidence to claim a higher mpg on the imported cars.


Hypothesis Testing Involving Paired Observations

Independent samples are samples that are not related in any way.

Dependent samples are samples that are paired or related in some fashion. For example: If you wished to buy a car you would look at

the same car at two (or more) different dealerships and compare the prices.

If you wished to measure the effectiveness of a new diet you would weigh the dieters at the start and at the finish of the program.


Hypothesis Testing Involving Paired Observations

Use the following test when the samples are dependent:

where is the mean of the differences is the standard deviation of the

differences n is the number of pairs (differences)

dsd

ns

dt

d


EXAMPLE 3

An independent testing agency is comparing the daily rental cost for renting a compact car from Hertz and Avis. A random sample of eight cities revealed the following information. At the .05 significance level can the testing agency conclude that there is a difference in the rental charged?


EXAMPLE 3 continued

City Hertz ($) Avis ($)

Atlanta 42 40

Chicago 56 52

Cleveland 45 43

Denver 48 48

Honolulu 37 32

Kansas City 45 48

Miami 41 39

Seattle 46 50


EXAMPLE 3 continued


H0: µd = 0 ; H1: µd ≠ 0


Step 3: Find the appropriate test statistic. We can use t as the test statistic.


EXAMPLE 3 continued

Step 4: State the decision rule. H0 is rejected if t < -2.365 or t > 2.365. We use the t distribution with 7 degrees of freedom.

365.22/ t

H0: µB ≤ µK ;

H1: µB > µK

Rejection Region IIprobability=0.025


Rejection Region IProbability =0.025

365.22/ t

Probability density of t statistic : t (df=7)


Example 3 continued

City Hertz ($) Avis ($) d d2

Atlanta 42 40 2 4

Chicago 56 52 4 16

Cleveland 45 43 2 4

Denver 48 48 0 0

Honolulu 37 32 5 25

Kansas City 45 48 -3 9

Miami 41 39 2 4

Seattle 46 50 -4 16


Example 3 continued

00.18

0.8

n

dd

1623.3

1888

78

1

222

nnd

dsd

894.081623.3

00.1

ns

dt

d


Example 3 continued

Step 5: Because 0.894 is less than the critical value, do not reject the null hypothesis. There is no difference in the mean amount charged by Hertz and Avis.

365.22/ t

H0: µB ≤ µK ;

H1: µB > µK

Rejection Region IIprobability=0.025


Rejection Region IProbability =0.025

365.22/ t

0.894


Two Sample Tests of Proportions

We investigate whether two samples came from populations with an equal proportion of successes.

The two samples are pooled using the following formula.

where X1 and X2 refer to the number of successes in the respective samples of n1 and n2.

21

21

nn

XXpc


Two Sample Tests of Proportions continued

The value of the test statistic is computed from the following formula.

21

21

)1()1(npp

npp

ppz

cccc


Example 4

Are unmarried workers more likely to be absent from work than married workers? A sample of 250 married workers showed 22 missed more than 5 days last year, while a sample of 300 unmarried workers showed 35 missed more than five days. Use a .05 significance level.


Example 4 continued

The null and the alternate hypothesis are:

H0: U ≤ M H1: U > M

The null hypothesis is rejected if the computed value of z is greater than 1.65.


Example 4 continued

The pooled proportion is

1036.250300

2235

cp

The value of the test statistic is

10.1

250)1036.1(1036.

300)1036.1(1036.25022

30035

z


Example 4 continued

The null hypothesis is not rejected. We cannot conclude that a higher proportion of unmarried workers miss more days in a year than the married workers.

The p-value is:P(z > 1.10) = .5000 - .3643 = .1457


- END -

Chapter ElevenTwo Sample Tests of Two Sample Tests of HypothesisHypothesis

dr. ka-fu wong

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