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*Perbandingan dua populasi Pertemuan 8 Matakuliah: D0722 - Statistika dan Aplikasinya Tahun: 2010*

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Perbandingan dua populasi Pertemuan 8Matakuliah: D0722 - Statistika dan AplikasinyaTahun: 2010

*Pada akhir pertemuan ini, diharapkan mahasiswa akan mampu :

membandingkan perbedaan antara dua nilai tengah populasi bebas dan berpasangan membandingkan perbedaan antara dua proporsi populasi dan dua ragam populasi Learning Outcomes

COMPLETE5 t h e d i t i o nBUSINESS STATISTICSAczel/SounderpandianMcGraw-Hill/Irwin The McGraw-Hill Companies, Inc., 20021-*

Using StatisticsPaired-Observation ComparisonsA Test for the Difference between Two Population Means Using Independent Random SamplesA Large-Sample Test for the Difference between Two Population ProportionsThe F Distribution and a Test for the Equality of Two Population VariancesSummary and Review of Terms

COMPLETE5 t h e d i t i o nBUSINESS STATISTICSAczel/SounderpandianMcGraw-Hill/Irwin The McGraw-Hill Companies, Inc., 20021-*

Inferences about differences between parameters of two populationsPaired-ObservationsObserve the same group of persons or thingsAt two different times: before and afterUnder two different sets of circumstances or treatmentsIndependent SamplesObserve different groups of persons or thingsAt different times or under different sets of circumstances8-1 Using Statistics

COMPLETE5 t h e d i t i o nBUSINESS STATISTICSAczel/SounderpandianMcGraw-Hill/Irwin The McGraw-Hill Companies, Inc., 20021-*

Population parameters may differ at two different times or under two different sets of circumstances or treatments because:The circumstances differ between times or treatmentsThe people or things in the different groups are themselves differentBy looking at paired-observations, we are able to minimize the between group , extraneous variation. Paired-Observation Comparisons

Paired-Observation Comparisons of Means

When paired data cannot be obtained, use independent random samples drawn at different times or under different circumstances.Large sample test if:Both n1 30 and n2 30 (Central Limit Theorem), orBoth populations are normal and 1 and 2 are both knownSmall sample test if:Both populations are normal and 1 and 2 are unknown A Test for the Difference between Two Population Means Using Independent Random Samples

I: Difference between two population means is 0 1= 2 H0: 1 -2 = 0H1: 1 -2 0II: Difference between two population means is less than 0 1 2H0: 1 -2 0H1: 1 -2 0III: Difference between two population means is less than D 1 2+DH0: 1 -2 DH1: 1 -2 DComparisons of Two Population Means: Testing Situations

Large-sample test statistic for the difference between two population means:

The term (1- 2)0 is the difference between 1 an 2 under the null hypothesis. Is is equal to zero in situations I and II, and it is equal to the prespecified value D in situation III. The term in the denominator is the standard deviation of the difference between the two sample means (it relies on the assumption that the two samples are independent).Comparisons of Two Population Means: Test Statistic

If we might assume that the population variances 12 and 22 are equal (even though unknown), then the two sample variances, s12 and s22, provide two separate estimators of the common population variance. Combining the two separate estimates into a pooled estimate should give us a better estimate than either sample variance by itself.

From both samples together we get a pooled estimate, sp2 , with (n1-1) + (n2-1) = (n1+ n2 -2) total degrees of freedom. A Test for the Difference between Two Population Means: Assuming Equal Population Variances

A pooled estimate of the common population variance, based on a sample variance s12 from a sample of size n1 and a sample variance s22 from a sample of size n2 is given by:

The degrees of freedom associated with this estimator is:df = (n1+ n2-2)The pooled estimate of the variance is a weighted average of the two individual sample variances, with weights proportional to the sizes of the two samples. That is, larger weight is given to the variance from the larger sample.Pooled Estimate of the Population Variance

Using the Pooled Estimate of the Population Variance

Hypothesized difference is zeroI: Difference between two population proportions is 0 p1= p2 H0: p1 -p2 = 0H1: p1 -p20II: Difference between two population proportions is less than 0 p1 p2H0: p1 -p2 0H1: p1 -p2 > 0Hypothesized difference is other than zero:III: Difference between two population proportions is less than D p1 p2+DH0:p-p2 DH1: p1 -p2 > D8-5 A Large-Sample Test for the Difference between Two Population Proportions

A large-sample test statistic for the difference between two population proportions, when the hypothesized difference is zero:

where is the sample proportion in sample 1 and is the sample

proportion in sample 2. The symbol stands for the combined sample proportion in both samples, considered as a single sample. That is:

When the population proportions are hypothesized to be equal, then a pooled estimator of the proportion ( ) may be used in calculating the test statistic. Comparisons of Two Population Proportions When the Hypothesized Difference Is Zero: Test Statistic

Carry out a two-tailed test of the equality of banks share of the car loan market in 1980 and 1995.Comparisons of Two Population Proportions When the Hypothesized Difference Is Zero: Example 8-8

Carry out a one-tailed test to determine whether the population proportion of travelers check buyers who buy at least $2500 in checks when sweepstakes prizes are offered as at least 10% higher than the proportion of such buyers when no sweepstakes are on.Comparisons of Two Population Proportions When the Hypothesized Difference Is Not Zero: Example 8-9

The F distribution is the distribution of the ratio of two chi-square random variables that are independent of each other, each of which is divided by its own degrees of freedom.An F random variable with k1 and k2 degrees of freedom:

The F Distribution and a Test for Equality of Two Population Variances

The F random variable cannot be negative, so it is bound by zero on the left.The F distribution is skewed to the right.The F distribution is identified the number of degrees of freedom in the numerator, k1, and the number of degrees of freedom in the denominator, k2.The F Distribution

Critical Points of the F Distribution Cutting Off a Right-Tail Area of 0.05

k1 1 2 3 4 5 6 7 8 9

k2 1161.4199.5215.7224.6230.2234.0236.8238.9240.5 218.5119.0019.1619.2519.3019.3319.3519.3719.38 310.139.559.289.129.018.948.898.858.81 47.716.946.596.396.266.166.096.046.00 56.615.795.415.195.054.954.884.824.77 65.995.144.764.534.394.284.214.154.10 75.594.744.354.123.973.873.793.733.68 85.324.464.073.843.693.583.503.443.39 95.124.263.863.633.483.373.293.233.18104.964.103.713.483.333.223.143.073.02114.843.983.593.363.203.09 3.01 2.952.90124.753.893.493.263.113.002.912.852.80134.673.813.413.183.032.922.832.772.71144.603.743.343.112.962.852.762.702.65154.543.683.293.062.902.792.712.642.59

3.015432100.70.60.50.40.30.20.10.0F0.05=3.01f(F)F Distribution with 7 and 11 Degrees of FreedomFThe left-hand critical point to go along with F(k1,k2) is given by:

Where F(k1,k2) is the right-hand critical point for an F random variable with the reverse number of degrees of freedom.Using the Table of the F Distribution

I: Two-Tailed Test 1 = 2 H0: 1 = 2 H1: 2 II: One-Tailed Test 12H0: 1 2 H1: 1 2Test Statistic for the Equality of Two Population Variances

The economist wants to test whether or not the event (interceptions and prosecution