Chapter Four Parameter Estimation and Statistical Inference

Statistics II_Chapter42 Sample and Sampling

Statistics II_Chapter43

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5 抽樣方法 n 簡單隨機抽樣 n 分層抽樣 n 部落抽樣 n 系統抽樣 統計之基礎理論與觀念

Statistics II_Chapter46 抽樣分配 n 中央極限定理 : 若母體為任意分配, 且母體之平均數為 m, 變異數為 s 2, 則 自母體抽取 n 個樣本, 若 n 夠大 (n>25), 若母體為任意分配, 且母體之平均數為 m, 變異數為 s 2, 則 自母體抽取 n 個樣本, 若 n 夠大 (n>25),樣本平均數樣本比例 n Examples 統計之基礎理論與觀念

Statistics II_Chapter47 Central Limit Theorem

Statistics II_Chapter48 Illustration of the Central Limit Theorem (Distribution of average scores from throwing dice)

Statistics II_Chapter49

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Statistics II_Chapter411

Statistics II_Chapter412 例、設某產品製程是常態分配 N(5,0.04), 抽樣 20 個產品資料, 試問 : (1) 這 20 個樣本平均數大於 5.02 的機率是多少 ? (2) 這 20 個樣本平均數介於 4.9 到 5.1 的機率是多少 ? (3) 這 20 個樣本總和大於 100 的機率是多少 ? (4) 這 20 個樣本總和大於 101 的機率是多少 ? (5) 這 20 個樣本平均數 x 之變異數是多少 ? (6) 這 20 個樣本總和之變異數是多少 ?

Statistics II_Chapter413 ( 續 ) (1) 利用中央極限定理求抽 50 件中樣本不良率 P 剛好為母體不良 率 1% 的機率 ? (2) 如果重複抽樣 400 次, 每次 50 個零件, 請描述樣本不良率 P 的分 佈狀況。 (3) 如果重複抽樣 400 次, 每次 100 個零件, 請描述不良率 P 分佈狀況。

Statistics II_Chapter414 例、若某製程已知不良率是 P=6%, 問 : (1) 抽樣 50 件, 則樣本不良率 P 與 P 相差在 1% 以內的機率是多少 ? (2) 抽樣 500 件, 則樣本不良率 P 與 P 相差在 1% 以內的機率是多少 ? (3) 抽樣 5000 件, 則樣本不良率 P 與 P 相差在 l% 以內的機率是多少 ?

Statistics II_Chapter415 Sample mean distribution Vs. 1. Population type 2. Sample size

Statistics II_Chapter416 點估計 (Point Estimation) n 以抽樣得來之樣本資料, 依循某一公式計算出單一數值, 來估計母體參數, 稱為點估計. n 好的點估計公式之條件 : 不偏性 不偏性 最小變異 最小變異 n 常用之點估計 : 母體平均數 ( m ) 母體平均數 ( m ) 母體變異數 ( s 2 ) 母體變異數 ( s 2 ) 統計之基礎理論與觀念

Statistics II_Chapter417

Statistics II_Chapter418 Criteria for Point Estimator n Unbiased n Minimum Variance n Absolute Efficiency n Relative Efficiency

Statistics II_Chapter419 不偏估計式 (Unbiased Estimator)

Statistics II_Chapter420

Statistics II_Chapter421

Statistics II_Chapter422 最小變異不偏估計式 Sample Mean X and X i are both unbiased estimator of , but the variance of sample mean (  2 /n) is less than the variance of X i (  2 ). Sample Mean X and X i are both unbiased estimator of , but the variance of sample mean (  2 /n) is less than the variance of X i (  2 ).

Statistics II_Chapter423 標準誤差 (Standard Error) n Used to measure the precision of estimation.

Statistics II_Chapter424

Statistics II_Chapter425 Absolute Efficiency 絕對有效性 n Used when no unbiased estimator are available. n Choose the estimator with smallest MSE.

Statistics II_Chapter426 Relative Efficiency 相對有效性 n Choose the estimator with relative smaller MSE.

Statistics II_Chapter427 Method of Maximum Likelihood 最大概似法

Statistics II_Chapter428

Statistics II_Chapter429 假設檢定 (Hypothesis Testing) n “A person is innocent until proven guilty beyond a reasonable doubt.” 在沒有充分證據證明其犯罪之前, 任何人皆是清白的. n 假設檢定 H0:  = 50 cm/s H1:   50 cm/s n Null Hypothesis (H 0 ) Vs. Alternative Hypothesis (H 1 ) n One-sided and two-sided Hypotheses n A statistical hypothesis is a statement about the parameters of one or more populations. 統計之基礎理論與觀念

Statistics II_Chapter430 About Testing n Critical Region n Acceptance Region n Critical Values

Statistics II_Chapter431 Errors in Hypothesis Testing n 檢定結果可能為 Type I Error(  ): Reject H 0 while H 0 is true. Type I Error(  ): Reject H 0 while H 0 is true. Type II Error(  ): Fail to reject H 0 while H 0 is false. Type II Error(  ): Fail to reject H 0 while H 0 is false. 統計之基礎理論與觀念

Statistics II_Chapter432

Statistics II_Chapter433 Making Conclusions We always know the risk of rejecting H 0, i.e.,  the significant level or the risk. We always know the risk of rejecting H 0, i.e.,  the significant level or the risk. We therefore do not know the probability of committing a type II error (  ). We therefore do not know the probability of committing a type II error (  ). n Two ways of making conclusion: 1. Reject H 0 2. Fail to reject H 0, (Do not say accept H 0 ) or there is not enough evidence to reject H 0. or there is not enough evidence to reject H 0. 統計之基礎理論與觀念

Statistics II_Chapter434 Significant Level (  )  = P(type I error) = P(reject H 0 while H 0 is true)  = P(type I error) = P(reject H 0 while H 0 is true) n = 10,  = 2.5  /  n = 0.79

Statistics II_Chapter435

Statistics II_Chapter436

Statistics II_Chapter437

Statistics II_Chapter438

Statistics II_Chapter439 The Power of a Statistical Test Power = 1 -  Power = 1 -  n Power = the sensitivity of a statistical test

Statistics II_Chapter From the problem context, identify the parameter of interest. 2. State the null hypothesis, H Specify an appropriate alternative hypothesis, H Choose a significance level a. 5. State an appropriate test statistic. 6. State the rejection region for the statistic. 7. Compute any necessary sample quantities, substitute these into the equation for the test statistic, and compute that value. 8. Decide whether or not H 0 should be rejected and report that in the problem context. General Procedure for Hypothesis Testing