Estimasi Parameter TIP-FTP-UB

Statistika Inferensial Terbagi dua bagian: Estimasi (estimation) Uji hipotesis (test of hypotheses)

Estimasi Terbagi menjadi dua bagian: Estimasi titik Estimasi interval

Point & Interval Estimation… Keller: Stats for Mgmt & Econ, 7th Ed April 14, 2017 Point & Interval Estimation… For example, suppose we want to estimate the mean summer income of a class of business students. For n=25 students, mean income is calculated to be 400 $/week. point estimate interval estimate An alternative statement is: The mean income is between 380 and 420 $/week. Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.

Estimator Qualities desirable in estimators: Unbiased An unbiased estimator of a population parameter is an estimator whose expected value is equal to that parameter. Jadi jika 𝜃 adalah parameter dan 𝜃 adalah estimator unbiased dari parameter 𝜃 apabila dipenuhi 𝐸 𝜃 =𝜃. Consistent An unbiased estimator is said to be consistent if the difference between the estimator and the parameter grows smaller as the sample size grows larger. Relatively efficient If there are two unbiased estimators of a parameter, the one whose variance is smaller is said to be relatively efficient.

Estimasi titik Sebuah nilai tunggal yang digunakan untuk mengestimasi sebuah parameter disebut titik estimator (atau cukup estimator), sedangkan proses untuk mengestimasi titik disebut estimasi titik (point estimation).

Estimasi Titik Dapat dibuktikan bahwa 𝑋 adalah estimator tak bias dari 𝜇 dan 𝑠 2 adalah estimator tak bias dari 𝜎 2 .

Estimasi Interval Proses untuk melakukan estimasi dengan menggunakan interval disebut estimasi interval. Derajat kepercayaan dalam mengestimasi disebut koefisien konfidensi. Misalnya 𝜃 merupakan estimator untuk parameter 𝜃, sedangkan A dan B adalah nilai-nilai estimator tersebut berdasarkan sampel tertentu, maka koefisien kepercayaannya dinyatakan dengan: 𝑃 𝐴<𝜃<𝐵 =1−𝛼 diartikan bahwa kita merasa 100(1−𝛼)% percaya (yakin) bahwa 𝜃 terletak diantara A dan B. 𝐴<𝜃<𝐵 disebut interval konfidensi (atau selang konfidensi), sedangkan A dan B disebut batas-batas kepercayaan. A disebut lower confidence limit, B disebut upper confidence limit

Interval Konfidensi untuk Rataan 𝜇 Untuk 𝜎 2 diketahui Jika 𝑋 adalah rataan sampel random berukuran n yang diambil dari populasi normal (atau populasi tak normal dengan ukuran sampel n≥30) dengan 𝜎 2 diketahui, maka interval konfidensi 100 1−𝛼 1−𝛼 % bagi 𝜇 ditentukan oleh: 𝑿 − 𝒛 𝜶 𝟐 𝝈 𝒏 <𝝁< 𝑿 + 𝒛 𝜶 𝟐 𝝈 𝒏 Untuk 𝜎 2 tak diketahui Jika 𝑋 dan 𝑠 2 berturut-turut adalah rataan dan variansi dari sampel random berukuran kecil (n<30) yang diambil dari populasi taknormal dengan 𝜎 2 tak diketahu, maka interval konfidensi 100 1−𝛼 % bagi 𝜇 ditentukan oleh: 𝑿 − 𝒛 𝜶 𝟐 𝒔 𝒏 <𝝁< 𝑿 + 𝒛 𝜶 𝟐 𝒔 𝒏

Four commonly used confidence levels… Keller: Stats for Mgmt & Econ, 7th Ed April 14, 2017 Four commonly used confidence levels… Confidence Level cut & keep handy!   Table 10.1 Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.

Keller: Stats for Mgmt & Econ, 7th Ed April 14, 2017 Example 1 A computer company samples demand during lead time over 25 time periods: It is known that the standard deviation of demand over lead time is 75 computers. We want to estimate the mean demand over lead time with 95% confidence in order to set inventory levels… Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.

Keller: Stats for Mgmt & Econ, 7th Ed April 14, 2017 Example 1 “We want to estimate the mean demand over lead time with 95% confidence in order to set inventory levels” Thus, the parameter to be estimated in the population is mean. And so our 𝜇 confidence interval estimator will be: IDENTIFY Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.

Keller: Stats for Mgmt & Econ, 7th Ed April 14, 2017 Example 1 In order to use our confidence interval estimator, we need the following pieces of data: therefore: The lower and upper confidence limits are 340.76 and 399.56. 370.16 1.96 75 n 25 Calculated from the data… Given Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.

Keller: Stats for Mgmt & Econ, 7th Ed April 14, 2017 Interval Width… A wide interval provides little information. For example, suppose we estimate with 95% confidence that an accountant’s average starting salary is between $15,000 and $100,000. Contrast this with: a 95% confidence interval estimate of starting salaries between $42,000 and $45,000. The second estimate is much narrower, providing accounting students more precise information about starting salaries. Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.

Keller: Stats for Mgmt & Econ, 7th Ed April 14, 2017 Interval Width… The width of the confidence interval estimate is a function of the confidence level, the population standard deviation, and the sample size… Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.

Keller: Stats for Mgmt & Econ, 7th Ed April 14, 2017 Interval Width The width of the confidence interval estimate is a function of the confidence level, the population standard deviation, and the sample size A larger confidence level produces a w i d e r confidence interval: Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.

Keller: Stats for Mgmt & Econ, 7th Ed April 14, 2017 Interval Width… The width of the confidence interval estimate is a function of the confidence level, the population standard deviation, and the sample size… Larger values of produce w i d e r confidence intervals Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.

Keller: Stats for Mgmt & Econ, 7th Ed April 14, 2017 Interval Width The width of the confidence interval estimate is a function of the confidence level, the population standard deviation, and the sample size Increasing the sample size decreases the width of the confidence interval while the confidence level can remain unchanged. Note: this also increases the cost of obtaining additional data Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.

Selecting the Sample Size Keller: Stats for Mgmt & Econ, 7th Ed April 14, 2017 Selecting the Sample Size We can control the width of the interval by determining the sample size necessary to produce narrow intervals. Suppose we want to estimate the mean demand “to within 5 units”; i.e. we want to the interval estimate to be: Since: It follows that Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.

Selecting the Sample Size Keller: Stats for Mgmt & Econ, 7th Ed April 14, 2017 Selecting the Sample Size Solving the equation that is, to produce a 95% confidence interval estimate of the mean (±5 units), we need to sample 865 lead time periods (vs. the 25 data points we have currently). Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.

Sample Size to Estimate a Mean Keller: Stats for Mgmt & Econ, 7th Ed April 14, 2017 Sample Size to Estimate a Mean The general formula for the sample size needed to estimate a population mean with an interval estimate of: Requires a sample size of at least this large: Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.

Keller: Stats for Mgmt & Econ, 7th Ed April 14, 2017 Example 2 A lumber company must estimate the mean diameter of trees to determine whether or not there is sufficient lumber to harvest an area of forest. They need to estimate this to within 1 inch at a confidence level of 99%. The tree diameters are normally distributed with a standard deviation of 6 inches. How many trees need to be sampled? Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.

Keller: Stats for Mgmt & Econ, 7th Ed April 14, 2017 Example 2 Things we know: Confidence level = 99%, therefore =.01 We want , hence W=1. We are given that = 6. 1 Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.

Keller: Stats for Mgmt & Econ, 7th Ed April 14, 2017 Example 2 We compute That is, we will need to sample at least 239 trees to have a 99% confidence interval of 1 Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.