1 Pertemuan 14 Peubah Acak Normal Matakuliah: I0134-Metode Statistika Tahun: 2007

2 Outline Materi: Sebaran rata-rata sampling Sebaran proporsi sampling

3 Sampling Distribution Theoretical Probability Distribution of a Sample Statistic Sample Statistic is a Random Variable –Sample mean, sample proportion Results from Taking All Possible Samples of the Same Size

4 Developing Sampling Distributions Suppose There is a Population … Population Size N=4 Random Variable, X, is Age of Individuals Values of X: 18, 20, 22, 24 Measured in Years A B C D

A B C D (18) (20) (22) (24) Uniform Distribution P(X) X Developing Sampling Distributions (continued) Summary Measures for the Population Distribution

6 All Possible Samples of Size n=2 16 Samples Taken with Replacement 16 Sample Means Developing Sampling Distributions (continued)

7 Sampling Distribution of All Sample Means X Sample Means Distribution 16 Sample Means _ Developing Sampling Distributions (continued)

8 Summary Measures of Sampling Distribution Developing Sampling Distributions (continued)

9 Comparing the Population with Its Sampling Distribution X Sample Means Distribution n = 2 A B C D (18) (20) (22) (24) Population N = 4 X _

10 Properties of Summary Measures –I.e., is unbiased Standard Error (Standard Deviation) of the Sampling Distribution is Less Than the Standard Error of Other Unbiased Estimators For Sampling with Replacement or without Replacement from Large or Infinite Populations: –As n increases, decreases

11 Unbiasedness ( ) BiasedUnbiased

12 Less Variability Sampling Distribution of Median Sampling Distribution of Mean Standard Error (Standard Deviation) of the Sampling Distribution is Less Than the Standard Error of Other Unbiased Estimators

13 Effect of Large Sample Larger sample size Smaller sample size

14 When the Population is Normal Central Tendency Variation Population Distribution Sampling Distributions

15 When the Population is Not Normal Central Tendency Variation Population Distribution Sampling Distributions

16 Central Limit Theorem As Sample Size Gets Large Enough Sampling Distribution Becomes Almost Normal Regardless of Shape of Population

17 How Large is Large Enough? For Most Distributions, n>30 For Fairly Symmetric Distributions, n>15 For Normal Distribution, the Sampling Distribution of the Mean is Always Normally Distributed Regardless of the Sample Size –This is a property of sampling from a normal population distribution and is NOT a result of the central limit theorem

18 Example: Sampling Distribution Standardized Normal Distribution

19 Population Proportions Categorical Variable –E.g., Gender, Voted for Bush, College Degree Proportion of Population Having a Characteristic Sample Proportion Provides an Estimate – If Two Outcomes, X Has a Binomial Distribution –Possess or do not possess characteristic

20 Sampling Distribution of Sample Proportion Approximated by Normal Distribution – –Mean: –Standard error: p = population proportion Sampling Distribution f(p s ) psps

21 Standardizing Sampling Distribution of Proportion Sampling Distribution Standardized Normal Distribution

22 Example: Sampling Distribution Standardized Normal Distribution