1 Pertemuan 16 Pendugaan Parameter Matakuliah: I0134 – Metoda Statistika Tahun: 2005 Versi: Revisi

2 Learning Outcomes Pada akhir pertemuan ini, diharapkan mahasiswa akan mampu : Mahasiswa dapat menghitung penduga selang dari rataan, proporsi dan varians.

3 Outline Materi Selang nilai tengah (rataan) Selang beda nilai tengah (rataan) Selang proporsi dan beda proporsi Selang varians dan proporsi varians

4 [ ]  Interval Estimation Interval Estimation of a Population Mean: Large-Sample Case Interval Estimation of a Population Mean: Small-Sample Case Determining the Sample Size Interval Estimation of a Population Proportion

5 Interval Estimation of a Population Mean: Large-Sample Case Sampling Error Probability Statements about the Sampling Error Constructing an Interval Estimate: Large-Sample Case with  Known Calculating an Interval Estimate: Large-Sample Case with  Unknown

6 Sampling Error The absolute value of the difference between an unbiased point estimate and the population parameter it estimates is called the sampling error. For the case of a sample mean estimating a population mean, the sampling error is Sampling Error =

7 Interval Estimate of a Population Mean: Large-Sample Case (n > 30) With  Known where: is the sample mean 1 -  is the confidence coefficient z  /2 is the z value providing an area of  /2 in the upper tail of the standard normal probability distribution  is the population standard deviation n is the sample size

8 Interval Estimate of a Population Mean: Large-Sample Case (n > 30) With  Unknown In most applications the value of the population standard deviation is unknown. We simply use the value of the sample standard deviation, s, as the point estimate of the population standard deviation.

9 Interval Estimation of a Population Mean: Small-Sample Case (n < 30) with  Unknown Interval Estimate where 1 -  = the confidence coefficient t  /2 = the t value providing an area of  /2 in the upper tail of a t distribution with n - 1 degrees of freedom s = the sample standard deviation

10 Contoh Soal: Apartment Rents Interval Estimation of a Population Mean: Small-Sample Case (n < 30) with  Unknown A reporter for a student newspaper is writing an article on the cost of off-campus housing. A sample of 10 one-bedroom units within a half-mile of campus resulted in a sample mean of $550 per month and a sample standard deviation of $60. Let us provide a 95% confidence interval estimate of the mean rent per month for the population of one-bedroom units within a half- mile of campus. We’ll assume this population to be normally distributed.

11 t Value At 95% confidence, 1 -  =.95,  =.05, and  /2 =.025. t.025 is based on n - 1 = = 9 degrees of freedom. In the t distribution table we see that t.025 = Contoh Soal: Apartment Rents

12 Estimation of the Difference Between the Means of Two Populations: Independent Samples Point Estimator of the Difference between the Means of Two Populations Sampling Distribution Interval Estimate of      Large-Sample Case Interval Estimate of      Small-Sample Case

13 Properties of the Sampling Distribution of –Expected Value –Standard Deviation where:  1 = standard deviation of population 1  2 = standard deviation of population 2 n 1 = sample size from population 1 n 2 = sample size from population 2 Sampling Distribution of

14 Interval Estimate with  1 and  2 Known where: 1 -  is the confidence coefficient Interval Estimate with  1 and  2 Unknown where: Interval Estimate of  1 -  2 : Large-Sample Case (n 1 > 30 and n 2 > 30)

15 95% Confidence Interval Estimate of the Difference Between Two Population Means: Large-Sample Case,  1 and  2 Unknown Substituting the sample standard deviations for the population standard deviation: = or yards to yards. We are 95% confident that the difference between the mean driving distances of Par, Inc. balls and Rap, Ltd. balls lies in the interval of to yards. Contoh Soal: Par, Inc.

16 Interval Estimate of  1 -  2 : Small-Sample Case (n 1 < 30 and/or n 2 < 30) Interval Estimate with  2 Known where:

17 95% Confidence Interval Estimate of the Difference Between Two Population Means: Small-Sample Case = or.3 to 4.7 miles per gallon. We are 95% confident that the difference between the mean mpg ratings of the two car types is from.3 to 4.7 mpg (with the M car having the higher mpg). Contoh Soal: Specific Motors

18 Inferences About the Difference Between the Proportions of Two Populations Sampling Distribution of Interval Estimation of p 1 - p 2 Hypothesis Tests about p 1 - p 2

19 Expected Value Standard Deviation Distribution Form If the sample sizes are large (n 1 p 1, n 1 (1 - p 1 ), n 2 p 2, and n 2 (1 - p 2 ) are all greater than or equal to 5), the sampling distribution of can be approximated by a normal probability distribution. Sampling Distribution of

20 Interval Estimation of  2 Interval Estimate of a Population Variance where the    values are based on a chi- square distribution with n - 1 degrees of freedom and where 1 -  is the confidence coefficient.

21 Chi-Square Distribution With Tail Areas of % of the possible  2 values 95% of the possible  2 values 22 2 Interval Estimation of  2

22 Selamat Belajar Semoga Sukses.