IENG 486 Statistical Quality & Process Control 4/17/2017 IENG 486 - Lecture 06 Hypothesis Testing & Excel Lab 4/17/2017 IENG 486 Statistical Quality & Process Control (c) 2002 - 2009, D.H. Jensen & R. C. Wurl

IENG 486 Statistical Quality & Process Control 4/17/2017 Assignment: Preparation: Print Hypothesis Test Tables from Materials page Have this available in class …or exam! Reading: Chapter 4: 4.1.1 through 4.3.4; (skip 4.3.5); 4.3.6 through 4.4.3; (skip rest) HW 2: CH 4: # 1a,b; 5a,c; 9a,c,f; 11a,b,d,g; 17a,b; 18, 21a,c; 22* *uses Fig.4.7, p. 126 4/17/2017 IENG 486 Statistical Quality & Process Control (c) 2002 - 2009, D.H. Jensen & R. C. Wurl

Relationship with Hypothesis Tests Assuming that our process is Normally Distributed and centered at the mean, how far apart should our specification limits be to obtain 99. 5% yield? Proportion defective will be 1 – .995 = .005, and if the process is centered, half of those defectives will occur on the right tail (.0025), and half on the left tail. To get 1 – .0025 = 99.75% yield before the right tail requires the upper specification limit to be set at  + 2.81. 4/17/2017 TM 720: Statistical Process Control

TM 720: Statistical Process Control 4/17/2017 TM 720: Statistical Process Control

Relationship with Hypothesis Tests Assuming that our process is Normally Distributed and centered at the mean, how far apart should our specification limits be to obtain 99. 5% yield? Proportion defective will be 1 – .995 = .005, and if the process is centered, half of those defectives will occur on the right tail (.0025), and half on the left tail. To get 1 – .0025 = 99.75% yield before the right tail requires the upper specification limit to be set at + 2.81. By symmetry, the remaining .25% defective should occur at the left side, with the lower specification limit set at  – 2.81 If we specify our process in this manner and made a lot of parts, we would only produce bad parts .5% of the time. 4/17/2017 IENG 486 Statistical Quality & Process Control

IENG 486 Statistical Quality & Process Control Hypothesis Tests An Hypothesis is a guess about a situation, that can be tested and can be either true or false. The Null Hypothesis has a symbol H0, and is always the default situation that must be proven wrong beyond a reasonable doubt. The Alternative Hypothesis is denoted by the symbol HA and can be thought of as the opposite of the Null Hypothesis - it can also be either true or false, but it is always false when H0 is true and vice-versa. 4/17/2017 IENG 486 Statistical Quality & Process Control

Hypothesis Testing Errors Type I Errors occur when a test statistic leads us to reject the Null Hypothesis when the Null Hypothesis is true in reality. The chance of making a Type I Error is estimated by the parameter  (or level of significance), which quantifies the reasonable doubt. Type II Errors occur when a test statistic leads us to fail to reject the Null Hypothesis when the Null Hypothesis is actually false in reality. The probability of making a Type II Error is estimated by the parameter . 4/17/2017 IENG 486 Statistical Quality & Process Control

IENG 486 Statistical Quality & Process Control 4/17/2017 Testing Example Single Sample, Two-Sided t-Test: H0: µ = µ0 versus HA: µ ¹ µ0 Test Statistic: Critical Region: reject H0 if |t| > t/2,n-1 P-Value: 2 x P(X ³ |t|), where the random variable X has a t-distribution with n _ 1 degrees of freedom 4/17/2017 IENG 486 Statistical Quality & Process Control (c) 2002 - 2009, D.H. Jensen & R. C. Wurl

IENG 486 Statistical Quality & Process Control 4/17/2017 Hypothesis Testing H0: m = m 0 versus HA: m  m 0 P-value = P(X£-|t|) + P(X³|t|) tn-1 distribution Critical Region: if our test statistic value falls into the region (shown in orange), we reject H0 and accept HA -|t| |t| 4/17/2017 IENG 486 Statistical Quality & Process Control (c) 2002 - 2009, D.H. Jensen & R. C. Wurl

Types of Hypothesis Tests Hypothesis Tests & Rejection Criteria 0 0   2  2  θ0 θ0 θ θ θ θ0 θ0 θ 0 One-Sided Test Statistic < Rejection Criterion H0: θ ≥ θ0 HA: θ < θ0 Two-Sided Test Statistic < -½ Rejection Criterion or Statistic > +½ Rejection Criterion H0: θ = θ0 HA: θ ≠ θ0 One-Sided Test Statistic > Rejection Criterion H0: θ ≤ θ0 HA: θ > θ0 4/17/2017 IENG 486 Statistical Quality & Process Control

Hypothesis Testing Steps State the null hypothesis (H0) from one of the alternatives: that the test statistic q = q0 , q ≥ q0 , or q ≤ q0 . Choose the alternative hypothesis (HA) from the alternatives: q ¹ q0 , q < q0 , or q > q0 . (Respectively!) Choose a significance level of the test (a). Select the appropriate test statistic and establish a critical region (q0). (If the decision is to be based on a P-value, it is not necessary to have a critical region) Compute the value of the test statistic () from the sample data. Decision: Reject H0 if the test statistic has a value in the critical region (or if the computed P-value is less than or equal to the desired significance level a); otherwise, do not reject H0. 4/17/2017 IENG 486 Statistical Quality & Process Control

IENG 486 Statistical Quality & Process Control 4/17/2017 Hypothesis Testing Significance Level of a Hypothesis Test: A hypothesis test with a significance level or size  rejects the null hypothesis H0 if a p-value smaller than is obtained, and accepts the null hypothesis H0 if a p-value larger than  is obtained. In this case, the probability of a Type I error (the probability of rejecting the null hypothesis when it is true) is equal to . True Situation Test Conclusion CORRECT Type I Error () H0 is False Type II Error () H0 is True 4/17/2017 IENG 486 Statistical Quality & Process Control (c) 2002 - 2009, D.H. Jensen & R. C. Wurl

IENG 486 Statistical Quality & Process Control 4/17/2017 Hypothesis Testing P-Value: One way to think of the P-value for a particular H0 is: given the observed data set, what is the probability of obtaining this data set or worse when the null hypothesis is true. A “worse” data set is one which is less similar to the distribution for the null hypothesis. P-Value 0.01 0.10 1 H0 not plausible Intermediate area H0 plausible 4/17/2017 IENG 486 Statistical Quality & Process Control (c) 2002 - 2009, D.H. Jensen & R. C. Wurl

Statistics and Sampling IENG 486 Statistical Quality & Process Control 4/17/2017 Statistics and Sampling Objective of statistical inference: Draw conclusions/make decisions about a population based on a sample selected from the population Random sample – a sample, x1, x2, …, xn , selected so that observations are independently and identically distributed (iid). Statistic – function of the sample data Quantities computed from observations in sample and used to make statistical inferences e.g. measures central tendency 4/17/2017 IENG 486 Statistical Quality & Process Control (c) 2002 - 2009, D.H. Jensen & R. C. Wurl

Sampling Distribution IENG 486 Statistical Quality & Process Control 4/17/2017 Sampling Distribution Sampling Distribution – Probability distribution of a statistic If we know the distribution of the population from which sample was taken, we can often determine the distribution of various statistics computed from a sample 4/17/2017 IENG 486 Statistical Quality & Process Control (c) 2002 - 2009, D.H. Jensen & R. C. Wurl

e.g. Sampling Distribution of the Average from the Normal Distribution IENG 486 Statistical Quality & Process Control 4/17/2017 e.g. Sampling Distribution of the Average from the Normal Distribution Take a random sample, x1, x2, …, xn, from a normal population with mean and standard deviation , i.e., Compute the sample average Then will be normally distributed with mean and std deviation That is 4/17/2017 IENG 486 Statistical Quality & Process Control (c) 2002 - 2009, D.H. Jensen & R. C. Wurl

Ex. Sampling Distribution of x IENG 486 Statistical Quality & Process Control 4/17/2017 Ex. Sampling Distribution of x When a process is operating properly, the mean density of a liquid is 10 with standard deviation 5. Five observations are taken and the average density is 15. What is the distribution of the sample average? r.v. x = density of liquid Ans: since the samples come from a normal distribution, and are added together in the process of computing the mean: 4/17/2017 IENG 486 Statistical Quality & Process Control (c) 2002 - 2009, D.H. Jensen & R. C. Wurl

Ex. Sampling Distribution of x (cont'd) IENG 486 Statistical Quality & Process Control 4/17/2017 Ex. Sampling Distribution of x (cont'd) What is the probability the sample average is greater than 15? Would you conclude the process is operating properly? 4/17/2017 IENG 486 Statistical Quality & Process Control (c) 2002 - 2009, D.H. Jensen & R. C. Wurl

IENG 486 Statistical Quality & Process Control 4/17/2017 IENG 486 Statistical Quality & Process Control

Ex. Sampling Distribution of x (cont'd) IENG 486 Statistical Quality & Process Control 4/17/2017 Ex. Sampling Distribution of x (cont'd) What is the probability the sample average is greater than 15? Would you conclude the process is operating properly? 4/17/2017 IENG 486 Statistical Quality & Process Control (c) 2002 - 2009, D.H. Jensen & R. C. Wurl