11.Hypothesis Testing II

Chapter 11

Hypothesis Testing IIStatistics for Business and Economics 6th Edition

Statistics for Business and Economics, 6e 2007 Pearson Education, Inc.

Setelah menyelesaikan bab ini, anda akan mampu : Melakukan uji hipotesis untuk perbedaan dua rata-rata populasiData berpasanganPopulasi Independen, variansi populasi diketahui Populasi Independen, variansi populasi tak diketahui Melakukan uji hipotesis dua proporsi (sampel besar )Use the F table to find critical F valuesComplete an F test for the equality of two variancesChapter Goals

Two Sample TestsTwo Sample TestsPopulation Means, Independent SamplesPopulation Means, Matched PairsPopulation VariancesGroup 1 vs. independent Group 2Same group before vs. after treatment Variance 1 vs.Variance 2Examples:Population ProportionsProportion 1 vs. Proportion 2 (Note similarities to Chapter 9)

Tests Means of 2 Related PopulationsPaired or matched samplesRepeated measures (before/after)Use difference between paired values:

Compute the average and standard dev of diAssumptions:Both Populations Are Normally DistributedMatched Pairsdi = xi - yi

The test statistic for the mean difference is a t value, with n 1 degrees of freedom:Test Statistic: Matched PairsWhereD0 = hypothesized mean differencesd = sample standard dev. of differencesn = the sample size (number of pairs)

Decision Rules: Matched PairsLower-tail test:

H0: x y 0H1: x y < 0Upper-tail test:

H0: x y 0H1: x y > 0Two-tail test:

H0: x y = 0H1: x y 0Paired Samplesaa/2a/2a-ta-ta/2tata/2Reject H0 if t < -tn-1, aReject H0 if t > tn-1, aReject H0 if t < -tn-1 , a/2 or t > tn-1 , a/2 Where has n - 1 d.f.

Assume you send your salespeople to a customer service training workshop. Has the training made a difference in the number of complaints? You collect the following data:

Matched Pairs Example Complaints: S.person Before (1) After (2) di

1 6 4 - 2 2 20 6 -14 3 3 2 - 1 4 0 0 0 5 4 0 - 4 -21 d =di

n = - 4.2


Has the training made a difference in the number of complaints? - 4.2d =H0: x y = 0H1: x y 0Test Statistic:Critical Value = 4.604 d.f. = n - 1 = 4Reject/2 - 4.604 4.604Decision: Do not reject H0(t stat is not in the reject region)Conclusion: There is not a significant change in the number of complaints.Matched Pairs: Solution Reject/2 - 1.66 = .01


Sumber data berbeda, pop 1 dan pop 2 Tidak berhubungan atau berpasanganIndependenSampel yang diambil dari satu populasi tidak berpengaruh terhadap sampel dari populasi lain

Difference Between Two MeansTujuan: Menguji perbedaan dua rata-rata populasi independent, x y

Difference Between Two MeansPopulation means, independent samplesTest statistic is a z valueTest statistic is a a value from the Students t distributionx2 and y2 assumed equalx2 and y2 knownx2 and y2 unknown x2 and y2 assumed unequal

x2 and y2 KnownPopulation means, independent samplesAssumptions:

Samples are randomly and independently drawn

both population distributions are normal

Population variances are known*x2 and y2 knownx2 and y2 unknown

x2 and y2 KnownPopulation means, independent samplesand the statistic

has a standard normal distributionWhen x2 and y2 are known and both populations are normal, the variance of X Y is*x2 and y2 knownx2 and y2 unknown

Hypothesis Tests forTwo Population MeansLower-tail test:

H0: x yH1: x < y

i.e.,


H0: x yH1: x > y

i.e.,


H0: x = yH1: x y

i.e.,

H0: x y = 0H1: x y 0Two Population Means, Independent Samples

Decision Rules Two Population Means, Independent Samples, Variances KnownLower-tail test:



H0: x y = 0H1: x y 0aa/2a/2a-za-za/2zaza/2Reject H0 if z < -zaReject H0 if z > zaReject H0 if z < -za/2 or z > za/2

x2 and y2 Unknown,Assumed EqualPopulation means, independent samplesAssumptions:


Populations are normally distributed

Population variances are unknown but assumed equal

*x2 and y2 assumed equalx2 and y2 knownx2 and y2 unknown x2 and y2 assumed unequal

x2 and y2 Unknown,Assumed EqualPopulation means, independent samplesTest Hypotheses:

The population variances are assumed equal, so use the two sample standard deviations and pool them to estimate

use a t value with (nx + ny 2) degrees of freedom*x2 and y2 assumed equalx2 and y2 knownx2 and y2 unknown x2 and y2 assumed unequal

Test Statistic, x2 and y2 Unknown, Equal*x2 and y2 assumed equalx2 and y2 unknown x2 and y2 assumed unequalWhere t has (n1 + n2 2) d.f.,andThe test statistic for x y is:

x2 and y2 Unknown,Assumed UnequalPopulation means, independent samplesAssumptions:


Populations are normally distributed

Population variances are unknown and assumed unequal

*x2 and y2 assumed equalx2 and y2 knownx2 and y2 unknown x2 and y2 assumed unequal

x2 and y2 Unknown,Assumed UnequalPopulation means, independent samplesForming interval estimates:

The population variances are assumed unequal, so a pooled variance is not appropriate

use a t value with degrees of freedom, wherex2 and y2 knownx2 and y2 unknown *x2 and y2 assumed equalx2 and y2 assumed unequal

Test Statistic, x2 and y2 Unknown, Unequal*x2 and y2 assumed equalx2 and y2 unknown x2 and y2 assumed unequalWhere t has degrees of freedom:The test statistic for x y is:

Decision RulesLower-tail test:



H0: x y = 0H1: x y 0aa/2a/2a-ta-ta/2tata/2Reject H0 if t < -tn-1, aReject H0 if t > tn-1, aReject H0 if t < -tn-1 , a/2 or t > tn-1 , a/2 Where t has n - 1 d.f.Two Population Means, Independent Samples, Variances Unknown

You are a financial analyst for a brokerage firm. Is there a difference in dividend yield between stocks listed on the NYSE & NASDAQ? You collect the following data: NYSE NASDAQ Number 21 25Sample mean 3.27 2.53Sample std dev 1.30 1.16Pooled Variance t Test: ExampleAssuming both populations are approximately normal with equal variances, is there a difference in average yield ( = 0.05)?

Calculating the Test StatisticThe test statistic is:

H0: 1 - 2 = 0 i.e. (1 = 2)H1: 1 - 2 0 i.e. (1 2) = 0.05df = 21 + 25 - 2 = 44Critical Values: t = 2.0154

Test Statistic:SolutionDecision:

Conclusion:

Reject H0 at a = 0.05There is evidence of a difference in means.t0 2.0154-2.0154.025Reject H0Reject H0.0252.040

Two Population ProportionsGoal: Test hypotheses for the difference between two population proportions, Px Py Population proportionsAssumptions: Both sample sizes are large, nP(1 P) > 9

The random variable

is approximately normally distributedTwo Population ProportionsPopulation proportions

Statistics for Business and Economics, 6e 2007 Pearson Education, Inc.Chap 11-*Test Statistic forTwo Population ProportionsPopulation proportionsThe test statistic for H0: Px Py = 0 is a z value: Where


Decision Rules: ProportionsPopulation proportionsLower-tail test:

H0: px py 0H1: px py < 0Upper-tail test:

H0: px py 0H1: px py > 0Two-tail test:

H0: px py = 0H1: px py 0aa/2a/2a-za-za/2zaza/2Reject H0 if z < -zaReject H0 if z > zaReject H0 if z < -za/2 or z > za/2

Is there a significant difference between the proportion of men and the proportion of women who will vote Yes on Proposition A?

In a random sample, 36 of 72 men and 31 of 50 women indicated they would vote Yes

Test at the .05 level of significanceExample: Two Population Proportions

The hypothesis test is:Statistics for Business and Economics, 6e 2007 Pearson Education, Inc.Chap 11-*

H0: PM PW = 0 (the two proportions are equal)H1: PM PW 0 (there is a significant difference betweenproportions)The sample proportions are:Men: = 36/72 = .50Women: = 31/50 = .62 The estimate for the common overall proportion is:Example: Two Population Proportions(continued)


Example: Two Population ProportionsThe test statistic for PM PW = 0 is:.025-1.961.96.025-1.31Decision: Do not reject H0Conclusion: There is not significant evidence of a difference between men and women in proportions who will vote yes.Reject H0Reject H0Critical Values = 1.96For = .05

Hypothesis Tests for Two VariancesTests for TwoPopulation VariancesF test statisticH0: x2 = y2H1: x2 y2Two-tail testLower-tail testUpper-tail testH0: x2 y2H1: x2 < y2H0: x2 y2H1: x2 > y2 Goal: Test hypotheses about two population variances The two populations are assumed to be independent and normally distributed

Hypothesis Tests for Two VariancesTests for TwoPopulation VariancesF test statisticThe random variableHas an F distribution with (nx 1) numerator degrees of freedom and (ny 1) denominator degrees of freedomDenote an F value with 1 numerator and 2 denominator degrees of freedom by

Test StatisticTests for TwoPopulation VariancesF test statisticThe critical value for a hypothesis test about two population variances iswhere F has (nx 1) numerator degrees of freedom and (ny 1) denominator degrees of freedom

Decision Rules: Two Variances rejection region for a two-tail test is:F 0 Reject H0Do not reject H0F 0 /2Reject H0Do not reject H0H0: x2 = y2H1: x2 y2H0: x2 y2H1: x2 > y2Use sx2 to denote the larger variance.where sx2 is the larger of the two sample variances

You are a financial analyst for a brokerage firm. You want to compare dividend yields between stocks listed on the NYSE & NASDAQ. You collect the following data: NYSE NASDAQ Number 2125Mean3.272.53Std dev1.301.16

Is there a difference in the variances between the NYSE & NASDAQ at the = 0.10 level?Example: F Test

Form the hypothesis test:H0: x2 = y2 (there is no difference between variances)H1: x2 y2 (there is a difference between variances)F Test: Example Solution Degrees of Freedom:Numerator (NYSE has the larger standard deviation):nx 1 = 21 1 = 20 d.f.Denominator: ny 1 = 25 1 = 24 d.f. Find the F critical values for = .10/2:

The test statistic is:F Test: Example Solution/2 = .05Reject H0Do not reject H0H0: x2 = y2H1: x2 y2F = 1.256 is not in the rejection region, so we do not reject H0Conclusion: There is not sufficient evidence of a difference in variances at = .10

F

For paired samples (t test):Tools | data analysis | t-test: paired two sample for means

For independent samples:Independent sample Z test with variances known:Tools | data analysis | z-test: two sample for means

For variancesF test for two variances:Tools | data analysis | F-test: two sample for variances

Two-Sample Tests in EXCEL

Two-Sample Tests in PHStat

Sample PHStat OutputInputOutput

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11.Hypothesis Testing II