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Pertemuan 16 Pengujian Hipotesis Proporsi dan Varians

Pertemuan 16 Pengujian Hipotesis Proporsi dan Varians. Matakuliah : I0284 - Statistika Tahun : 2005 Versi : Revisi. Learning Outcomes. Pada akhir pertemuan ini, diharapkan mahasiswa akan mampu : Mahasiswa akan dapat menyusun simpulan dari hasil uji hipotesis proporsi dan varians.

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Pertemuan 16 Pengujian Hipotesis Proporsi dan Varians

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  1. Pertemuan 16Pengujian Hipotesis Proporsi dan Varians Matakuliah : I0284 - Statistika Tahun : 2005 Versi : Revisi

  2. Learning Outcomes Pada akhir pertemuan ini, diharapkan mahasiswa akan mampu : • Mahasiswa akan dapat menyusun simpulan dari hasil uji hipotesis proporsi dan varians.

  3. Outline Materi • Uji hipotesis proporsi sampel besar • Uji hipotesis beda proporsi sampel besar • Uji hipotesis varians • Uji hipotesis kesamaan dua varians

  4. Hypothesis Testing • Developing Null and Alternative Hypotheses • Type I and Type II Errors • One-Tailed Tests About a Population Mean: Large-Sample Case • Two-Tailed Tests About a Population Mean: Large-Sample Case • Tests About a Population Mean: Small-Sample Case continued

  5. Hypothesis Testing • Tests About a Population Proportion • Hypothesis Testing and Decision Making • Calculating the Probability of Type II Errors • Determining the Sample Size for a Hypothesis Test about a Population Mean

  6. Developing Null and Alternative Hypotheses • Hypothesis testing can be used to determine whether a statement about the value of a population parameter should or should not be rejected. • The null hypothesis, denoted by H0 , is a tentative assumption about a population parameter. • The alternative hypothesis, denoted by Ha, is the opposite of what is stated in the null hypothesis. • Hypothesis testing is similar to a criminal trial. The hypotheses are: H0: The defendant is innocent Ha: The defendant is guilty

  7. Developing Null and Alternative Hypotheses • Testing Research Hypotheses • The research hypothesis should be expressed as the alternative hypothesis. • The conclusion that the research hypothesis is true comes from sample data that contradict the null hypothesis.

  8. A Summary of Forms for Null and Alternative Hypotheses about a Population Mean • The equality part of the hypotheses always appears in the null hypothesis. • In general, a hypothesis test about the value of a population mean  must take one of the following three forms (where 0 is the hypothesized value of the population mean). H0: >0 H0: <0 H0:  = 0 Ha:  < 0 Ha:  > 0 Ha: 0

  9. Type I and Type II Errors • Since hypothesis tests are based on sample data, we must allow for the possibility of errors. • A Type I error is rejecting H0 when it is true. • A Type II error is accepting H0 when it is false. • The person conducting the hypothesis test specifies the maximum allowable probability of making a Type I error, denoted by  and called the level of significance. • Generally, we cannot control for the probability of making a Type II error, denoted by . • Statistician avoids the risk of making a Type II error by using “do not reject H0” and not “accept H0”.

  10. Contoh Soal: Metro EMS • Type I and Type II Errors Population Condition H0 True Ha True Conclusion ( ) ( ) Accept H0Correct Type II (Conclude  Conclusion Error RejectH0Type I Correct (Conclude  rror Conclusion

  11. The Use of p-Values • The p-value is the probability of obtaining a sample result that is at least as unlikely as what is observed. • The p-value can be used to make the decision in a hypothesis test by noting that: • if the p-value is less than the level of significance , the value of the test statistic is in the rejection region. • if the p-value is greater than or equal to , the value of the test statistic is not in the rejection region. • Reject H0 if the p-value < .

  12. The Steps of Hypothesis Testing • Determine the appropriate hypotheses. • Select the test statistic for deciding whether or not to reject the null hypothesis. • Specify the level of significance  for the test. • Use to develop the rule for rejecting H0. • Collect the sample data and compute the value of the test statistic. • a) Compare the test statistic to the critical value(s) in the rejection rule, or b) Compute the p-value based on the test statistic and compare it to to determine whether or not to reject H0.

  13. One-Tailed Tests about a Population Mean: Large-Sample Case (n> 30) • Hypotheses • H0:   or H0:  Ha: Ha: • Test Statistic  Known Unknown • Rejection Rule Reject H0 if z > zReject H0 if z < -z

  14. Contoh Soal: Metro EMS • One-Tailed Test about a Population Mean: Large n Let  = P(Type I Error) = .05 Sampling distribution of (assuming H0 is true and  = 12) Reject H0 Do Not Reject H0  1.645 c 12 (Critical value)

  15. Contoh Soal: Metro EMS • One-Tailed Test about a Population Mean: Large n Let n = 40, = 13.25 minutes, s = 3.2 minutes (The sample standard deviation s can be used to estimate the population standard deviation .) Since 2.47 > 1.645, we reject H0. Conclusion: We are 95% confident that Metro EMS is not meeting the response goal of 12 minutes; appropriate action should be taken to improve service.

  16. Contoh Soal: Metro EMS • Using the p-value to Test the Hypothesis Recall that z = 2.47 for = 13.25. Then p-value = .0068. Since p-value < , that is .0068 < .05, we reject H0. Reject H0 Do Not Reject H0 p-value z 0 1.645 2.47

  17. Two-Tailed Tests about a Population Mean: Large-Sample Case (n> 30) • Hypotheses H0: =  Ha:  • Test Statistic  Known Unknown • Rejection Rule Reject H0 if |z| > z

  18. Summary of Test Statistics to be Used in aHypothesis Test about a Population Mean Yes No n > 30 ? No Popul. approx. normal ? s known ? Yes Yes Use s to estimate s No s known ? No Use s to estimate s Yes Increase n to > 30

  19. A Summary of Forms for Null and Alternative Hypotheses about a Population Proportion • The equality part of the hypotheses always appears in the null hypothesis. • In general, a hypothesis test about the value of a population proportion pmust take one of the following three forms (where p0 is the hypothesized value of the population proportion). H0: p>p0H0: p<p0H0: p = p0 Ha: p < p0Ha: p > p0Ha: pp0

  20. Contoh Soal: NSC • Two-Tailed Test about a Population Proportion: Large n • Hypothesis H0: p = .5 Ha: p .5 • Test Statistic

  21. Hypothesis TestingAbout a Population Variance • Right-Tailed Test • Hypotheses • Test Statistic • Rejection Rule Reject H0 if (where is based on a chi-square distribution with n - 1 d.f.) or Reject H0 if p-value < a

  22. Hypothesis TestingAbout a Population Variance • Two-Tailed Test • Hypotheses • Test Statistic • Rejection Rule Reject H0 if (where are based on a chi-square distribu- tion with n - 1 d.f.) or Reject H0 if p-value < a

  23. Selamat Belajar Semoga Sukses.

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