Parameter Estimation and Statistical Inference

Parameter Estimation and Statistical Inference
Chapter Four Parameter Estimation and Statistical Inference

Statistics II_Chapter4
Sample and Sampling Horng-Chyi Horng Statistics II_Chapter4

Horng-Chyi Horng Statistics II_Chapter4

抽樣方法統計之基礎理論與觀念簡單隨機抽樣分層抽樣部落抽樣系統抽樣 Horng-Chyi Horng Statistics II_Chapter4

抽樣分配統計之基礎理論與觀念中央極限定理: 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), 樣本平均數樣本比例 Examples Horng-Chyi Horng Statistics II_Chapter4

Central Limit Theorem Horng-Chyi Horng Statistics II_Chapter4

Illustration of the Central Limit Theorem (Distribution of average scores from throwing dice) Horng-Chyi Horng Statistics II_Chapter4

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? Horng-Chyi Horng Statistics II_Chapter4

(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。 Horng-Chyi Horng Statistics II_Chapter4

(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? Horng-Chyi Horng Statistics II_Chapter4

Sample mean distribution Vs. 1. Population type 2. Sample size Horng-Chyi Horng Statistics II_Chapter4

點估計(Point Estimation)
統計之基礎理論與觀念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. 好的點估計公式之條件: 不偏性最小變異常用之點估計: 母體平均數(m) 母體變異數(s 2) Horng-Chyi Horng Statistics II_Chapter4

Criteria for Point Estimator
Unbiased Minimum Variance Absolute Efficiency Relative Efficiency Horng-Chyi Horng Statistics II_Chapter4

不偏估計式(Unbiased Estimator) Horng-Chyi Horng Statistics II_Chapter4

最小變異不偏估計式 Sample Mean X and Xi are both unbiased estimator of m, but the variance of sample mean (s2/n) is less than the variance of Xi (s2). Horng-Chyi Horng Statistics II_Chapter4

標準誤差(Standard Error) Used to measure the precision of estimation. Horng-Chyi Horng Statistics II_Chapter4

Absolute Efficiency 絕對有效性
Used when no unbiased estimator are available. Choose the estimator with smallest MSE. Horng-Chyi Horng Statistics II_Chapter4

Relative Efficiency 相對有效性 Choose the estimator with relative smaller MSE. Horng-Chyi Horng Statistics II_Chapter4

Method of Maximum Likelihood 最大概似法

假設檢定(Hypothesis Testing)
統計之基礎理論與觀念 “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. 假設檢定 H0: m = 50 cm/s H1: m  50 cm/s Null Hypothesis (H0) Vs. Alternative Hypothesis (H1) One-sided and two-sided Hypotheses A statistical hypothesis is a statement about the parameters of one or more populations. Horng-Chyi Horng Statistics II_Chapter4

About Testing Critical Region Acceptance Region Critical Values Horng-Chyi Horng Statistics II_Chapter4

Errors in Hypothesis Testing 統計之基礎理論與觀念檢定結果可能為 Type I Error(a): Reject H0 while H0 is true. Type II Error(b): Fail to reject H0 while H0 is false. Horng-Chyi Horng Statistics II_Chapter4

Making Conclusions 統計之基礎理論與觀念 We always know the risk of rejecting H0, i.e., a, the significant level or the risk. We therefore do not know the probability of committing a type II error (b). Two ways of making conclusion: 1. Reject H0 2. Fail to reject H0, (Do not say accept H0) or there is not enough evidence to reject H0. Horng-Chyi Horng Statistics II_Chapter4

Significant Level (a) a = P(type I error) = P(reject H0 while H0 is true) n = 10, s = 2.5 s/n = 0.79 Horng-Chyi Horng Statistics II_Chapter4

The Power of a Statistical Test
Power = 1 - b Power = the sensitivity of a statistical test Horng-Chyi Horng Statistics II_Chapter4

General Procedure for Hypothesis Testing 1. From the problem context, identify the parameter of interest. 2. State the null hypothesis, H0. 3. Specify an appropriate alternative hypothesis, H1. 4. 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 H0 should be rejected and report that in the problem context. Horng-Chyi Horng Statistics II_Chapter4

Exercise 8-1 A Textile fiber manufacturer is investigating a new drapery yarn, which the company claims has a mean thread elongation of 12 kg with a standard deviation of 0.5kg. The company wishes to test the hypothesis H0: m = 12 against H1: m < 12, using a random sample of four specimens. What is the type I error probability if the critical region is defined as x < 11.5 kg? Find b for the case where the true mean elongation is 11.25kg. Horng-Chyi Horng Statistics II_Chapter4

Exercise 8-2, 8-3 8-2 Repeat Exercise 8-1 using a sample size of n = 16 and the same critical region. 8-3 In Exercise 8-1, find the boundary of the critical region if the type I error probability is specified to be a = 0.01. Horng-Chyi Horng Statistics II_Chapter4

Parameter Estimation and Statistical Inference

Similar presentations

Presentation on theme: "Parameter Estimation and Statistical Inference"— Presentation transcript:

Similar presentations

About project

反馈

请登录

Auth with social network:

Parameter Estimation and Statistical Inference

Similar presentations

Presentation on theme: "Parameter Estimation and Statistical Inference"— Presentation transcript:

Similar presentations

About project

反馈