1 Pertemuan 15 Pendugaan Parameter Nilai Tengah Matakuliah: I0134 – Metode Statistika Tahun: 2007

2 Learning Outcomes Pada akhir pertemuan ini, diharapkan mahasiswa akan mampu : Mahasiswa akan dapat menghitung pendugaan parameter nilai tengah satu atau dua populasi.

3 Outline Materi Penduigaan nilai tengah satu populasi Pendugaan beda dua nilai tengah sampel besar Pendugaan beda nilai tengah sampel kecil Pendugaan beda nilai tengah populasi tidak bebas

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 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

6 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.

7 Interval Estimation of a Population Mean: Small-Sample Case (n < 30) Population is Not Normally Distributed The only option is to increase the sample size to n > 30 and use the large-sample interval-estimation procedures. Population is Normally Distributed and  is Known The large-sample interval-estimation procedure can be used. Population is Normally Distributed and  is Unknown The appropriate interval estimate is based on a probability distribution known as the t distribution.

8 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

9 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)

10 Point Estimator of the Difference Between the Means of Two Populations Population 1 Par, Inc. Golf Balls  1 = mean driving distance of Par distance of Par golf balls Population 1 Par, Inc. Golf Balls  1 = mean driving distance of Par distance of Par golf balls Population 2 Rap, Ltd. Golf Balls  2 = mean driving distance of Rap distance of Rap golf balls Population 2 Rap, Ltd. Golf Balls  2 = mean driving distance of Rap distance of Rap golf balls m 1 –  2 = difference between the mean distances the mean distances Simple random sample Simple random sample of n 1 Par golf balls of n 1 Par golf balls x 1 = sample mean distance for sample of Par golf ball Simple random sample Simple random sample of n 1 Par golf balls of n 1 Par golf balls x 1 = sample mean distance for sample of Par golf ball Simple random sample Simple random sample of n 2 Rap golf balls of n 2 Rap golf balls x 2 = sample mean distance for sample of Rap golf ball Simple random sample Simple random sample of n 2 Rap golf balls of n 2 Rap golf balls x 2 = sample mean distance for sample of Rap golf ball x 1 - x 2 = Point Estimate of m 1 –  2

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

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

13 Point Estimate of the Difference Between Two Population Means  1 = mean miles-per-gallon for the population of M cars  2 = mean miles-per-gallon for the population of J cars Point estimate of  1 -  2 = = = 2.5 mpg. Contoh Soal: Specific Motors

14 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

15 Inference About the Difference Between the Means of Two Populations: Matched Samples With a matched-sample design each sampled item provides a pair of data values. The matched-sample design can be referred to as blocking. This design often leads to a smaller sampling error than the independent-sample design because variation between sampled items is eliminated as a source of sampling error.

16 Delivery Time (Hours) District OfficeUPX INTEX Difference Seattle Los Angeles Boston Cleveland New York Houston Atlanta St. Louis Milwaukee Denver Contoh Soal: Express Deliveries

17 Inference About the Difference Between the Means of Two Populations: Matched Samples Let  d = the mean of the difference values for the two delivery services for the population of district offices –Hypotheses H 0 :  d = 0, H a :  d  –Rejection Rule Assuming the population of difference values is approximately normally distributed, the t distribution with n - 1 degrees of freedom applies. With  =.05, t.025 = (9 degrees of freedom). Reject H 0 if t Contoh Soal: Express Deliveries

18 Inference About the Difference Between the Means of Two Populations: Matched Samples Conclusion Reject H 0. There is a significant difference between the mean delivery times for the two services. Contoh Soal: Express Deliveries

19 Selamat Belajar Semoga Sukses.