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Matakuliah Tahun Versi : I0014 / Biostatistika : 2005 : V1 / R1 Pertemuan 11 Pengujian Hipotesis (I) 1 Learning Outcomes Pada akhir pertemuan ini, diharapkan mahasiswa akan mampu : • Mahasiswa dapat menjelaskan konsep pengujian hipotesis (C2) • Mahasiswa dapat menguji hipotesis untuk nilai tengah (C3) 2 Outline Materi • Pendugaan Nilai tengah ( ) • Pendugaan beda dua nilai tengah (1 2 ) 3 <<ISI>> Pengujian Hipotesis • • A null hypothesis, denoted by H0, is an assertion about one or more population parameters. This is the assertion we hold to be true until we have sufficient statistical evidence to conclude otherwise. – H0: =100 The alternative hypothesis, denoted by H1, is the assertion of all situations not covered by the null hypothesis. – H1: 100 • H0 and H1 are: – Mutually exclusive – Only one can be true. – Exhaustive – Together they cover all possibilities, so one or the other must be true. 4 <<ISI>> Logika Pengujian Hipotesis A contingency table illustrates the possible outcomes of a statistical hypothesis test. 5 <<ISI>> Kesalahan dalam Uji Hipotesis • A decision may be incorrect in two ways: – Type I Error: Reject a true H0 • • The Probability of a Type I error is denoted by . is called the level of significance of the test – Type II Error: Accept a false H0 • • • The Probability of a Type II error is denoted by . 1 - is called the power of the test. and are conditional probabilities: = P(Reject H 0 H 0 is true) = P(Accept H 0 H 0 is false) 6 <<ISI>> Pengujian Mean Populasi (n besar) Null Hypothesis H0: = 0 Alternative Hypothesis H0: 0 Significance Level of the Test often 0.05 or 0.01) Test Statistic z Critical Points Critical Points of z z 2 x 0 (assuming is unknown, s otherwise substitute for s) n The bounds z that capture an area of (1-) 2 Decision Rule Reject the null hypothesis if either z > z a or z < -z a 2 2 0.01 0.02 0.05 0.10 0.20 0.005 0.010 0.025 0.050 0.100 2 2.576 2.326 1.960 1.645 1.282 7 <<ISI>> Pengujian Mean Populasi (n kecil) When the population is normal, the population standard deviation,, is unknown and the sample size is small, the hypothesis test is based on the t distribution, with (n-1) degrees of freedom, rather than the standard normal distribution. Small - sample test statistic for the population mean, : x - 0 t= s n When the population is normally distributed and the null hypothesis is true, the test statistic has a t distribution with n -1 degrees of freedom 8 <<ISI>> Uji mean berpasangan (pair t test) Test statistic for the paired - observations t test: D D t sD n where D is the sample average difference between each pair of observations, sD is the sample standard deviation of these differences, and the sample size, n, is the number of pairs of observations. The symbol D is the population mean difference under the null hypothesis. When the null hypothesis is true and the population mean difference is D , the statistic has a t distribution with (n - 1) degrees of freedom. 0 0 0 9 <<ISI>> Uji Mean Dua Populasi Independen 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), or • Both populations are normal and 1 and 2 are both known – Small sample test if: • Both populations are normal and 1 and 2 are unknown10 <<ISI>> Situasi Pengujian Dua Mean Populasi • I: Difference between two population means is 0 • H0: 1 -2 = 0 • H1: 1 -2 0 • II: Difference between two population means is less than 0 • H0: 1 -2 0 • H1: 1 -2 0 • III: Difference between two population means is less than D • H0: 1 -2 D • H1: 1 -2 D 11 <<ISI>> Statistik Uji Dua Mean Populasi Large-sample test statistic for the difference between two population means: ( x1 x2 ) ( 1 2 ) 0 z s12 s22 n1 n2 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). 12 <<ISI>> Uji Dua Mean Populasi dengan Ukuran Contoh Kecil • When sample sizes are small (n1< 30 or n2< 30 or both), and both populations are normally distributed, the test statistic • has approximately a t distribution with degrees of freedom given by (round downward to the nearest integer if necessary): t ( x1 x2 ) ( 1 2 )0 s12 s22 n1 n2 2 2 s22 s1 n1 n2 df 2 2 2 2 s1 s2 n1 n 2 n1 1 n2 1 13 <<ISI>> Menggunakan Ragam gabungan (Pooled Variance) The estimate of the standard deviation of (x1 x 2 ) is given by: 1 2 1 sp n1 n 2 Test statistic for the difference between two population means, assuming equal population variances: (x1 x 2 ) ( 1 2 ) 0 t= 1 2 1 sp n1 n2 where ( 1 2 ) 0 is the difference between the two population means under the null hypothesis (zero or some other number D). The number of degrees of freedom of the test statistic is df = ( n1 n2 2 ) (the 2 number of degrees of freedom associated with s p , the pooled estimate of the population variance. 14 << CLOSING>> • Sampai saat ini Anda telah mempelajari pengujian hipotesis nilai tengah, baik untuk satu populasi maupun dua populasi • Untuk dapat lebih memahami penggunaan pengujian hipotesis tersebut, cobalah Anda pelajari materi penunjang, dan mengerjakan latihan 15

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