Chapter 4 Statistical Inference in Quality Control and Improvement. Statistical Quality Control (D. C. Montgomery)

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1 Chapter 4 Statistical Inference in Quality Control and Improvement 許湘伶 Statistical Quality Control (D. C. Montgomery)

2 Sampling distribution I a random sample of size n: if it is selected so that the observations {x i } are i.i.d. x: central tendency s 2 : variability

3 Sampling from a Normal Distribution I x N (µ, σ 2 ) x 1,..., x n is sampled from N (µ, σ 2 ) x = x 1,..., x n i.i.d N (0, 1) ni=1 x i n N (µ, σ2 n ) n y = xi 2 χ 2 n i=1 (Γ( n 2, 2)) f (y) = yn/2 1 e y/2 Γ(n/2)2 n/2, y > 0 E(y) = n Var(y) = 2n

4 Sampling from a Normal Distribution II x 1,..., x n N (µ, σ 2 ) ni=1 (x i x) 2 ( y = = σ 2 ) (n 1)s2 σ 2 χ 2 n 1

5 Sampling from a Normal Distribution III x N (0, 1), y χ 2 k and x y t = x y k t k f (t) = ( ) Γ ((k + 1)/2) t 2 (k+1)/2 kπγ(k/2) k + 1, < t < E(t) = 0, Var(t) = k k 2, for k > 2

6 Sampling from a Normal Distribution IV k the t distribution is closely approximate N (0, 1) Generally k > 30 x 1,..., x n N (µ, σ 2 ) x N (µ, σ 2 n ) (n 1)s2 σ 2 χ 2 n 1 x s 2 x µ s/ n t n 1

7 Sampling from a Normal Distribution V w χ 2 u, y χ 2 v, and w y x F u,v ( f (x) = Γ u+v 2 Γ ( ) ( u 2 Γ v ) 2 F u,v = w/u y/v F u,v ( u v ) u/2 ) x u/2 1 [( u v ) x + 1 ] (u+v)/2 0 < x <

8 Sampling from a Normal Distribution VI x 11,..., x 1n1 N (µ 1, σ 2 1 ), x 21,..., x 2n2 N (µ 2, σ 2 2 ) (n i 1)s 2 i σ 2 i χ ni 1, i = 1, 2 s 2 1 /σ2 1 s 2 2 /σ2 2 F n1 1,n 2 1

9 Sampling from a Bernoulli Distribution I x Ber(p) (p: the probability of success) x 1,..., x n Ber(p) p(x) = { p, x = 1 (1 p), x = 0 n x = x i B(n, p) i=1 E( x) = p, Var( x) = p(1 p) n

10 Sampling from a Poisson Distribution I x 1,..., x m P(λ) x = n x i P(nλ) i=1 E( x) = λ, Var( x) = λ n Linear combinations of Poisson r.v. are used in quality engineering work x i i.i.d P(λ i ), i = 1,..., m, {a i } : constants m different types of defects The distribution of L = m i=1 a i x i is not Poission unless all a i =1

11 Point Estimation of Process Parameters I Statistical quality control: the probability distribution is used to describe or model some critical-to-quality( 關鍵質量要素 ) control The parameters of the probability distributions are generally unknown A point estimator: a statistic that produces a single numerical value as the estimate of the unknown parameter Distribution Population parameter sample estimate N (µ, σ 2 ) µ x N (µ, σ 2 ) σ 2 s 2 P(λ) λ x B(n, p) p (n: fixed) x

12 Point Estimation of Process Parameters II Properties Important properties are required of good point estimators: 1. Unbiased: E(ˆθ) = θ 2. Minimum variance E( x) = µ Var(s 2 ) = σ 2 E(s) = c 4 σ = ( ) 1/2 ( ) 2 Γ(n/2) n 1 Γ((n 2)/2) σ E s c 4 = σ

13 Point Estimation of Process Parameters III Range method: to estimate the standard deviation x 1..., x n N (µ, σ 2 ) The range sample: R = max 1 i n (x i) min 1 i n (x i) = xmax x min The relative range: W = R σ E(W ) = d 2 (d 2 : Appendix Table VI (2 n 25)) ˆσ = R d 2

14 Statistical Inference for a Single Sample I Techniques of statistical inference: parameter estimation hypothesis testing Statistical hypothesis: a statement about the values of the parameters of a probability distribution Testing a hypothesis Null hypothesis (H 0 ) and alternative hypothesis (H 1 ) Testing statistics Type of errors Critical region (or rejection region) Reject of fail to reject the H 0 Confidence interval

15 Statistical Inference for a Single Sample II Type of errors Hypothesis H 0 is true H 1 is true Fail to reject H 0 Correct Decision Type II error (β) Reject H 0 Type I error (α) Correct Decision (1 β=power) α = P{type I error} = P{reject H 0 H 0 is true} (producer s risk) β = P{type II error} = P{fail to rejecth 0 H 0 is false} (consumer s risk) Power = 1 β = P{reject H 0 H 0 is false}

16 Statistical Inference for a Single Sample III Figure : The type of errors (obtained from The Errors of A/B Testing)

17 Statistical Inference for a Single Sample IV P-value conveys( 傳達 ) much information about the weight of evidence against H 0 The P-value is the smallest level of significance that would lead to rejection of H 0. Figure : The illustration of P-value (obtained from Wiki)

18 Statistical Inference for a Single Sample V Confidence interval (C.I.): Ex: Two-sided 100(1 α)% C.I. construct an interval estimator of the mean µ P{L µ U } = 1 α 100(1 α)% C.I. of µ: L µ U One-sided lower 100(1 α)% C.I. : P{L µ} = 1 α L µ One-sided upper 100(1 α)% C.I. : P{µ U } = 1 α µ U

19 Statistical Inference for a Single Sample VI Hypothesis testing for a single sample: Inference on µ: variance known or variance unknown Inference on σ 2 Inference on p (population proportion) Sample size decisions

20 Statistical Inference for a Single Sample VII Inference Sided Hypothesis Testing statistics Critical region 100(1 α)% C.I. Z 0 > Z α/2 σ x Z α/2 n σ µ x + Z α/2 n µ(σ known) µ(σ unknown) Two-sided H 0 : µ = µ 0 H 1 : µ µ 0 One-sided Z 0 = x µ0 σ/ n H 0 : µ = µ 0 Z 0 > Z α x Z α/2 H 1 : µ > µ 0 σ n µ H 0 : µ = µ 0 Z 0 < Z α x + Z α/2 σ n µ H 1 : µ < µ 0 Two-sided H 0 : µ = µ 0 H 1 : µ µ 0 One-sided t 0 = x µ0 s/ n t 0 > t α/2,n 1 x t α/2,n 1 s n µ x + t α/2,n 1 s n H 0 : µ = µ 0 t 0 > t α,n 1 x t α,n 1 H 1 : µ > µ 0 s n µ H 0 : µ = µ 0 t 0 < t α,n 1 x + t α,n 1 s n µ H 1 : µ < µ 0 z N (0, 1) P(z>Z α/2 ) = α/2 t t n 1 P(t>t α/2,n 1 ) = α/2

21 Statistical Inference for a Single Sample VIII Test the hypothesis: the variance of a normal distribution equals a constant Inference Sided Hypothesis Testing statistics Critical region 100(1 α)% C.I. (n 1)s 2 σ 2 χ 2 α/2,n 1 σ Two-sided H 0 : σ = σ 0 H 1 : σ σ 0 One-sided χ 2 0 = (n 1)s2 σ 2 0 χ 2 0 > χ2 α/2,n 1 or χ 2 0 < χ2 1 α/2,n 1 H 0 : σ = σ 0 H 1 : σ > σ 0 χ 2 0 < χ2 1 α,n 1 σ 2 (n 1)s2 χ 2 1 α,n 1 H 0 : σ = σ 0 H 1 : σ < σ 0 χ 2 0 > χ2 α,n 1 (n 1)s 2 σ 2 χ 2 α,n 1 (n 1)s2 χ 2 1 α/2,n 1 P(χ 2 n 1 >χ2 α/2 ) = α/2 and P(χ2 n 1 >χ2 1 α/2 ) = 1 α/2

22 Statistical Inference for a Single Sample IX Test the hypothesis: the proportion p of a population equals a standard value (p 0 ) a random sample of n items is taken from the population x items in the sample belong to the class associated with p 二項分佈趨近至常態分佈 Inference Sided Hypothesis Testing statistics Critical 100(1 α)% C.I. region ˆp(1 ˆp) Two-sided H 0 : p = p 0 (x+0.5) np0 if x < np 0 Z 0 > Z α/2 ˆp Zα/2 p ˆp + n Zα/2 np0(1 p0) p H 1 : p p 0 Z 0 = (x 0.5) np0 if x > np 0 np0(1 p0) ˆp(1 ˆp) n ˆp = x n

23 Statistical Inference for a Single Sample X n is large, p 0.1 the normal approximation to the binomial can be used n is small the binomial distribution C.I. n is large, p is small the Poisson distribution approximation to the binomial is useful to constructing C.I.

24 Sample size Decisions I Probability of type II error H 0 : µ = µ 0 vs. H 1 : µ µ 0 Z 0 = x µ 0 σ/ n N (0, 1) Suppose µ 1 = µ + δ, δ > 0 (Under H 1 ) The type II error: β = Φ ( Z 0 = x µ 0 n σ/ n N (δ σ, 1) Z α/2 δ n σ ) Φ ( Z α/2 δ ) n σ

25 Sample size Decisions II β is a function of n, δ, α Specify α and design a test procedure maximize the power ( minimize β, a function of sample size.) Operating-characteristic (OC) curves( 操作特性曲線 ): d = δ /σ

26 Sample size Decisions III Figure : Operating-characteristic curves for two-sided normal test with α = 0.05 fixed n, α d β Power fixed δ, α n β Power

27 Sample size Decisions IV > library(pwr) > pwr.t.test(n=25,d=0.75,sig.level=.01,alternative="greater",type ="one.sample") One-sample t test power calculation n = 25 d = 0.75 sig.level = 0.01 power = alternative = greater > pwr.t.test(d=0.1/0.1,sig.level=0.05,power=0.85,type ="one.sample") One-sample t test power calculation n = d = 1 sig.level = 0.05 power = 0.85 alternative = two.sided

28 Inference for two samples I Inference for a difference in Means: variance known or variance unknown Inference on the variances of two normal distributions Inference on two population proportions

29 Inference for two samples II E( x 1 x 2 ) = µ 1 µ 2 Var( x 1 x 2 ) = σ2 1 n 1 + σ2 2 n 2 The quantity: Z = x 1 x 2 (µ 1 µ 2 ) N (0, 1) σ1 2 n 1 + σ2 2 n 2

30 Inference for two samples III

31 Inference for two samples IV Case 1: σ 2 1 = σ2 2 = σ2 when variances unknown E( x 1 x 2 ) = µ 1 ( µ 2 ) Var( x 1 x 2 ) = σ 2 1 n n 2 The pooled estimator of σ 2 s 2 p = (n1 1)s2 1 + (n 2 1)s 2 2 n 1 + n 2 2 t test statistic: = ws 2 1+(1 w)s 2 2 (weighted average) t = x 1 x 2 (µ 1 µ 2 ) s p 1 n n 2 t n1 +n 2 2

32 Inference for two samples V

33 Inference for two samples VI Case 2: σ 2 1 σ2 2 when variances unknown The test statistic: t0 = x 1 x 2 (µ 1 µ 2 ) t ν, s1 2 n 1 + s2 2 n 2 ( s 2 ) 2 1 n 1 + s2 2 n 2 ν = (s1 2/n 1) 2 n (s2 2 /n 2) 2 n 2 1 2

34 Inference for two samples VII Paired data d j = x 1j x 2j, j = 1,..., n Hypothesis: The t test statistic: H 0 : µ d = 0 H 0 : µ 1 = µ 2 t 0 = Reject H 0 if t 0 > t α/2,n 1 d s d / n t n 1

35 Inference for two samples VIII

36 Inference for two samples IX ˆp = x 1+x 2 n 1 +n 2

Chapter 2. Hypothesis testing in one population

Chapter 2. Hypothesis testing in one population Contents Introduction, the null and alternative hypotheses Hypothesis testing process Type I and Type II errors, power Test statistic, level of significance