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Matakuliah Tahun Versi : I0284 - Statistika : 2005 : Revisi Pertemuan 16 Pengujian Hipotesis Proporsi dan Varians 1 Learning Outcomes Pada akhir pertemuan ini, diharapkan mahasiswa akan mampu : • Mahasiswa akan dapat menyusun simpulan dari hasil uji hipotesis proporsi dan varians. 2 Outline Materi • • • • Uji hipotesis proporsi sampel besar Uji hipotesis beda proporsi sampel besar Uji hipotesis varians Uji hipotesis kesamaan dua varians 3 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 4 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 5 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 6 Ha: The defendant is guilty 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. 7 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 Ha: < 0 H0: < 0 Ha: > 0 H0: = 0 Ha: 0 8 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”. 9 Contoh Soal: Metro EMS • Type I and Type II Errors Conclusion Accept H0 (Conclude Reject H0 (Conclude Population Condition H0 True Ha True ( ) ( ) Correct Conclusion Type II Error Type I Correct rror Conclusion 10 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 < . 11 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. 12 One-Tailed Tests about a Population Mean: Large-Sample Case (n > 30) Hypotheses H0: Ha: Test Statistic Known z x 0 / n or H0: Ha: Unknown z x 0 s/ n Rejection Rule Reject H0 if z > zReject H0 if z < -z 13 Contoh Soal: Metro EMS • One-Tailed Test about a Population Mean: Large n Let = P(Type I Error) = .05 Sampling distribution of x (assuming H0 is true and = 12) Reject H0 Do Not Reject H0 1.645 x 12 x c (Critical value) 14 Contoh Soal: Metro EMS • One-Tailed Test about a Population Mean: Large n Let n = 40, x = 13.25 minutes, s = 3.2 minutes (The sample standard deviation s can be used to estimate the population standard deviation .) x 13. 25 12 z 2. 47 / n 3. 2 / 40 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. 15 Contoh Soal: Metro EMS • Using the p-value to Test the Hypothesis Recall that z = 2.47 for x = 13.25. Then p-value = .0068. Since p-value < , that is .0068 < .05, we reject H0. Reject H0 Do Not Reject H0 0 p-value 1.645 2.47 z 16 Two-Tailed Tests about a Population Mean: Large-Sample Case (n > 30) • Hypotheses H0: Ha: • Test Statistic Known Unknown x 0 x 0 z z / n s/ n • Rejection Rule Reject H0 if |z| > z 17 Summary of Test Statistics to be Used in a Hypothesis Test about a Population Mean Yes s known ? Yes n > 30 ? No Yes Use s to estimate s s known ? Yes z x / n No x z s/ n x z / n No Popul. approx. normal ? No Use s to estimate s x t s/ n Increase n to > 30 18 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 p must take one of the following three forms (where p0 is the hypothesized value of the population proportion). H0: p > p0 Ha: p < p0 H0: p < p0 Ha: p > p0 H0: p = p0 Ha: p p0 19 Contoh Soal: NSC • Two-Tailed Test about a Population Proportion: Large n – Hypothesis H0: p = .5 Ha: p .5 – Test Statistic p z p0 (1 p0 ) .5(1 .5) .045644 n 120 p p0 (67 /120) .5 1.278 p .045644 20 Hypothesis Testing About a Population Variance Right-Tailed Test • Hypotheses H 0 : 2 20 H a : 2 20 • Test Statistic 2 • Rejection Rule ( n 1) s 2 20 2 Reject H0 if (where (1 ) is based on a chi-square distribution with n - 1 d.f.) or Reject H0 if p-value < 2 2(1 ) 21 Hypothesis Testing About a Population Variance Two-Tailed Test • Hypotheses H 0 : 2 20 H a : 2 20 • Test Statistic • Rejection Rule 2 ( n 1) s 2 20 2 2(1 / 2) or 2 2 / 2 2 2 and Reject H0 if (1 /2) /2 (where are based on a chi-square distribution with n - 1 d.f.) or Reject H0 if p-value < 22 • Selamat Belajar Semoga Sukses. 23