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Chapter FourChapter Four
Parameter Estimation and Statistical InferenceParameter Estimation and Statistical Inference
Statistics II_Chapter4Statistics II_Chapter4 22
Sample and SamplingSample and Sampling
Statistics II_Chapter4Statistics II_Chapter4 33
Statistics II_Chapter4Statistics II_Chapter4 44
Statistics II_Chapter4Statistics II_Chapter4 55
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
Statistics II_Chapter4Statistics II_Chapter4 66
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
統計之基礎理論與觀念統計之基礎理論與觀念
),(~2
nNX
))1(
,(~n
PPPNP
Statistics II_Chapter4Statistics II_Chapter4 77
Central Limit TheoremCentral Limit Theorem
Statistics II_Chapter4Statistics II_Chapter4 88
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)
Statistics II_Chapter4Statistics II_Chapter4 99
Statistics II_Chapter4Statistics II_Chapter4 1010
Statistics II_Chapter4Statistics II_Chapter4 1111
Statistics II_Chapter4Statistics II_Chapter4 1212
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_Chapter4Statistics II_Chapter4 1313
( 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_Chapter4Statistics II_Chapter4 1414
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_Chapter4Statistics II_Chapter4 1515
Sample mean distributionVs. 1. Population type 2. Sample size
Statistics II_Chapter4Statistics II_Chapter4 1616
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 (())
統計之基礎理論與觀念統計之基礎理論與觀念
n
XX i
1
1
2
2
n
XXS
n
ii
Statistics II_Chapter4Statistics II_Chapter4 1717
Statistics II_Chapter4Statistics II_Chapter4 1818
Criteria for Point EstimatorCriteria for Point Estimator
UnbiasedUnbiased
Minimum VarianceMinimum Variance
Absolute EfficiencyAbsolute Efficiency
Relative EfficiencyRelative Efficiency
Statistics II_Chapter4Statistics II_Chapter4 1919
不偏估計式不偏估計式 (Unbiased Estimator)(Unbiased Estimator)
Statistics II_Chapter4Statistics II_Chapter4 2020
Statistics II_Chapter4Statistics II_Chapter4 2121
Statistics II_Chapter4Statistics II_Chapter4 2222
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).).
Statistics II_Chapter4Statistics II_Chapter4 2323
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
Statistics II_Chapter4Statistics II_Chapter4 2424
Statistics II_Chapter4Statistics II_Chapter4 2525
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
Statistics II_Chapter4Statistics II_Chapter4 2626
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
Statistics II_Chapter4Statistics II_Chapter4 2727
Method of Maximum LikelihoodMethod of Maximum Likelihood最大概似法最大概似法
Statistics II_Chapter4Statistics II_Chapter4 2828
Statistics II_Chapter4Statistics II_Chapter4 2929
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.
統計之基礎理論與觀念統計之基礎理論與觀念
Statistics II_Chapter4Statistics II_Chapter4 3030
About TestingAbout Testing
Critical RegionCritical Region Acceptance RegionAcceptance Region Critical ValuesCritical Values
Statistics II_Chapter4Statistics II_Chapter4 3131
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.
統計之基礎理論與觀念統計之基礎理論與觀念
Statistics II_Chapter4Statistics II_Chapter4 3232
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
Statistics II_Chapter4Statistics II_Chapter4 3333
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..
統計之基礎理論與觀念統計之基礎理論與觀念
Statistics II_Chapter4Statistics II_Chapter4 3434
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
Statistics II_Chapter4Statistics II_Chapter4 3535
Statistics II_Chapter4Statistics II_Chapter4 3636
Statistics II_Chapter4Statistics II_Chapter4 3737
Statistics II_Chapter4Statistics II_Chapter4 3838
Statistics II_Chapter4Statistics II_Chapter4 3939
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
Statistics II_Chapter4Statistics II_Chapter4 4040
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