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Ch 10 實習 Concepts of Estimation 2 The objective of estimation is to determine the value of a population parameter on the basis of a sample statistic. There are two types of estimators: Point Estimator Interval estimator Jia-Ying Chen Point Estimator A point estimator draws inference about a population by estimating the value of an unknown parameter using a single value or point. 3 Jia-Ying Chen Point Estimator A point estimator draws inference about a population by estimating the value of an unknown parameter using a single value or point. Parameter Population distribution ? Sampling distribution 4 Point estimator Jia-Ying Chen Interval Estimator An interval estimator draws inferences about a population by estimating the value of an unknown parameter using an interval. Population distribution Parameter Sample distribution Interval estimator 5 Jia-Ying Chen Estimator’s Characteristics Selecting the right sample statistic to estimate a parameter value depends on the characteristics of the statistic. Estimator’s desirable characteristics: Unbiasedness: An unbiased estimator is one whose expected value is equal to the parameter it estimates. Consistency: An unbiased estimator is said to be consistent if the difference between the estimator and the parameter grows smaller as the sample size increases. Relative efficiency: For two unbiased estimators, the one with a smaller variance is said to be relatively efficient. 6 Jia-Ying Chen Unbiased Estimate 估計的量,我們會希望它不要系統性的高估、或者系統 性的低估,也就是要求估計量有「不偏」性質。用秤體重來 比喻的話,如果體重計有時把我們秤重了些、有時又秤輕了 些,但是若秤了許許多多次之後,平均起來就等於我們的真 實體重的話,就相當於有不偏性質 E (X ) 7 E ( pˆ ) p 2 (x x) 2 s n 1 E (s 2 ) 2 Jia-Ying Chen Consistency… 8 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. E.g. X is a consistent estimator of because: V(X) is That is, as n grows larger, the variance of X grows smaller. Jia-Ying Chen Relative Efficiency… 9 If there are two unbiased estimators of a parameter, the one whose variance is smaller is said to be relatively efficient. E.g. both the the sample median and sample mean are unbiased estimators of the population mean, however, the sample median has a greater variance than the sample mean, so we choose since it is relatively efficient when compared to the sample median. Jia-Ying Chen Example 1 a. b. c. d. 10 A point estimate is defined as: The average of the sample values The average of the population values A single value that is the best estimate of an unknown population parameter A single value that is the best estimate of an unknown sample statistic Jia-Ying Chen Example 2 a. b. c. d. 11 Which of the following statements is correct? The sample mean is an unbiased estimator of the population mean The sample proportion is an unbiased estimator of the population proportion The difference between two sample means is an unbiased estimator of the difference between two population means All of the above Jia-Ying Chen Estimating the Population Mean when the Population Variance is Known How is an interval estimator produced from a sampling distribution? 12 A sample of size n is drawn from the population, and its mean x is calculated. By the central limit theorem x is normally distributed (or approximately normally distributed.), thus… Jia-Ying Chen Estimating the Population Mean when the Population Variance is Known x Z n We have established before that P( z 2 13 x z 2 ) 1 n n Jia-Ying Chen The Confidence Interval for ( is known) This leads to the following equivalent statement P( x z 2 x z 2 ) 1 n n The confidence interval 14 Jia-Ying Chen Interpreting the Confidence Interval for 1 – of all the values of x obtained in repeated sampling from a given distribution, construct an interval x z 2 n , x z 2 n that includes (covers) the expected value of the population. 15 Jia-Ying Chen Graphical Demonstration of the Confidence Interval for Confidence level 1- x z 2 Lower confidence limit 16 n x 2z 2 x z 2 n n Upper confidence limit Jia-Ying Chen The Confidence Interval for ( is known) Four commonly used confidence levels Confidence level 0.90 0.95 0.98 0.99 17 0.10 0.05 0.02 0.01 /2 0.05 0.025 0.01 0.005 z/2 1.645 1.96 2.33 2.575 Jia-Ying Chen The Width of the Confidence Interval 2z 2 n The width of the confidence interval is affected by • • • 18 The population standard deviation () The confidence level (1-) The sample size (n). Jia-Ying Chen Example 3 a. b. c. d. 19 In developing an interval estimate for a population mean, the population standard deviation σ was assumed to be 10. The interval estimate was 50.92±2.14. Had σ equaled 20, the interval estimate would be 60.92±2.14 50.92±12.14 101.84±4.28 50.92±4.28 Jia-Ying Chen Example 4 a. The mean of a random sample of 25 observations from a normal population whose standard deviation is 40 is 200. Estimate the population mean with 95% confidence. b. Repeat part a changing the population standard deviation to 25. c. Repeat part a changing the population standard deviation to 10 d. Describe what happens to the confidence interval estimate when the standard deviation is decreased 20 Jia-Ying Chen Solution 40 a. x z / 2 200 1.96( ) 200 15.68, LCL=184.32, UCL=215.68 n 25 25 b. x z / 2 200 1.96( ) 200 9.8, LCL=190.2, UCL=209.8 n 25 10 c. x z / 2 200 1.96( ) 200 3.92, LCL=196.08, UCL=203.92 n 25 d . The interval narrows 21 Jia-Ying Chen Example 5 22 The following data represent a random sample of 9 marks (out of 10) on a statistics quiz. The marks are normally distributed with a standard deviation of 2. Estimate the population mean with 90% confidence. 4 6 8 5 4 8 3 10 9 Jia-Ying Chen Solution 23 2 6.33 1.645( ) 6.33 1.10 n 9 x z / 2 LCL=5.23, UCL=7.43 Jia-Ying Chen Selecting the Sample size We can control the width of the confidence interval by changing the sample size. Thus, we determine the interval width first, and derive the required sample size. The phrase “estimate the mean to within W units”, translates to an interval estimate of the form xw 24 Jia-Ying Chen Selecting the Sample size The required sample size to estimate the mean is z 2 n w 25 2 Any non integer value must be rounded up. Jia-Ying Chen Example 6 a. A statistics practitioner would like to estimate a population mean to within 10 units. The confidence level has been set at 95% and =200. Determine the sample size b. Suppose that the sample mean was calculated as 500. Estimate the population mean with 95% confidence 26 Jia-Ying Chen Solution 27 a. n=(zα/2×σ/W)2=(1.96×200/10)2=1537 b. 500±10 Jia-Ying Chen Example 7 28 A statistics professor wants to compare today’s students with those 25 years ago. All of his current students’ marks are stored on a computer so that he can easily determine the population mean. However, the marks 25 years ago reside only in his musty files. He does not want to retrieve all the marks and will be satisfied with a 95% confidence interval estimate of the mean mark 25 years ago. If he assumes that the population standard deviation is 10, how large a sample should he take to estimate the mean to within 3 marks? Jia-Ying Chen Solution 29 n=(zα/2×σ/W)2=(1.96×10/3)2=43 Jia-Ying Chen