chapter four parameter estimation and statistical inference

Chapter FourChapter Four

Parameter Estimation and Statistical InferenceParameter Estimation and Statistical Inference

Statistics II_Chapter4Statistics II_Chapter4 22

Sample and SamplingSample and Sampling


抽樣方法抽樣方法簡單隨機抽樣簡單隨機抽樣

分層抽樣分層抽樣

部落抽樣部落抽樣

系統抽樣系統抽樣

統計之基礎理論與觀念統計之基礎理論與觀念


抽樣分配抽樣分配中央極限定理中央極限定理 :: 若母體為任意分配若母體為任意分配 , , 且母體之平均數為且母體之平均數為 , , 變異數為變異數為 ,,

則自母體抽取則自母體抽取 n n 個樣本個樣本 , , 若若 n n 夠大夠大 (n>25), (n>25),

樣本平均數樣本平均數

樣本比例樣本比例

ExamplesExamples


),(~2

nNX

))1(

,(~n

PPPNP


Central Limit TheoremCentral Limit Theorem


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


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 ?


( 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。


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 ?


Sample mean distributionVs. 1. Population type 2. Sample size


點估計點估計 (Point Estimation)(Point Estimation)

以抽樣得來之樣本資料以抽樣得來之樣本資料 , , 依循某一公式計算出單一數依循某一公式計算出單一數值值 , , 來估計母體參數來估計母體參數 , , 稱為點估計稱為點估計 ..

好的點估計公式之條件好的點估計公式之條件 ::• 不偏性不偏性• 最小變異最小變異

常用之點估計常用之點估計 ::• 母體平均數母體平均數 (())

• 母體變異數母體變異數 (())


n

XX i

1

1

2

2

n

XXS

n

ii


Criteria for Point EstimatorCriteria for Point Estimator

UnbiasedUnbiased

Minimum VarianceMinimum Variance

Absolute EfficiencyAbsolute Efficiency

Relative EfficiencyRelative Efficiency


不偏估計式不偏估計式 (Unbiased Estimator)(Unbiased Estimator)


最小變異不偏估計式最小變異不偏估計式

Sample Mean X and XSample Mean X and Xii are both unbiased estimator of are both unbiased estimator of , ,

but the variance of sample mean (but the variance of sample mean (22/n) is less than the vari/n) is less than the variance of Xance of Xi i ((22).).


標準誤差標準誤差 (Standard Error)(Standard Error)

Used to measure the precision of estimation.Used to measure the precision of estimation.

V


Absolute Efficiency Absolute Efficiency 絕對有效性絕對有效性

Used when no unbiased estimator are available.Used when no unbiased estimator are available. Choose the estimator with smallest MSE.Choose the estimator with smallest MSE.

2)()()( biasVMSE


Relative Efficiency Relative Efficiency 相對有效性相對有效性 Choose the estimator with relative smaller MSE.Choose the estimator with relative smaller MSE.

. choose , 1)(

)( 1

2

1

MSE

MSEIF


Method of Maximum LikelihoodMethod of Maximum Likelihood最大概似法最大概似法


假設檢定假設檢定 (Hypothesis Testing)(Hypothesis Testing)

““A person is innocent until proven guilty beyond a reasonA person is innocent until proven guilty beyond a reasonable doubt.” able doubt.” 在沒有充分證據證明其犯罪之前在沒有充分證據證明其犯罪之前 , , 任何任何人皆是清白的人皆是清白的 ..

假設檢定假設檢定H0: H0: = 50 cm/s = 50 cm/s

H1: H1: 50 cm/s 50 cm/s Null Hypothesis (HNull Hypothesis (H00) Vs. Alternative Hypothesis (H) Vs. Alternative Hypothesis (H11))

One-sided and two-sided HypothesesOne-sided and two-sided Hypotheses A statistical hypothesis is a statement about the parameterA statistical hypothesis is a statement about the parameter

s of one or more populations.s of one or more populations.



About TestingAbout Testing

Critical RegionCritical Region Acceptance RegionAcceptance Region Critical ValuesCritical Values


Errors in Hypothesis TestingErrors in Hypothesis Testing

檢定結果可能為檢定結果可能為

Type I Error(Type I Error(): Reject H): Reject H00 while H while H00 is true. is true.

Type II Error(Type II Error(): Fail to reject H): Fail to reject H00 while H while H00 is false. is false.



The Defendant isThe Jury finds the

person Innocent Guilty

Innocent Type II Error

Guilty Type I Error

)(:

)(:

1

0

GuiltyH

InnocentH

有罪無辜


Making ConclusionsMaking Conclusions

We always know the risk of rejecting HWe always know the risk of rejecting H00, i.e., , i.e., the the

significant level or the risk.significant level or the risk. We therefore do not know the probability of committing a We therefore do not know the probability of committing a

type II error (type II error ().).

Two ways of making conclusion:Two ways of making conclusion:

1. Reject H1. Reject H00

2. Fail to reject H2. Fail to reject H00, (Do not say accept H, (Do not say accept H00))

or there is not enough evidence to reject Hor there is not enough evidence to reject H00..



Significant Level (Significant Level ()) = P(type I error) = P(reject H= P(type I error) = P(reject H00 while H while H00 is true) is true)

n = 10, = 2.5/n = 0.79


The Power of a Statistical TestThe Power of a Statistical Test

Power = 1 - Power = 1 - Power = the sensitivity of a statistical testPower = the sensitivity of a statistical test


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.

General Procedure for Hypothesis General Procedure for Hypothesis TestingTesting

chapter four parameter estimation and statistical inference

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