IENG 486 - Lecture 06

1. IENG 486 - Lecture 06 Hypothesis Testing & Excel Lab IENG 486 Statistical Quality & Process Control

2. 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 IENG 486 Statistical Quality & Process Control

3. 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. TM 720: Statistical Process Control

4. TM 720: Statistical Process Control

5. 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. IENG 486 Statistical Quality & Process Control

6. Hypothesis Tests • AnHypothesis 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 wrongbeyond 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. IENG 486 Statistical Quality & Process Control

7. Hypothesis Testing Errors • Type I Errorsoccur 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 . IENG 486 Statistical Quality & Process Control

8. 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 IENG 486 Statistical Quality & Process Control

9. Hypothesis Testing H0: m = m0 versus HA: mm0 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 rejectH0and accept HA 0 |t| -|t| IENG 486 Statistical Quality & Process Control

10.  2  2   θ0 θ θ θ0 θ0 θ θ θ0 0 0 Types of Hypothesis Tests Hypothesis Tests & Rejection Criteria 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 IENG 486 Statistical Quality & Process Control

11. 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. IENG 486 Statistical Quality & Process Control

12. True Situation H0 is True H0 is False H0 is True CORRECT Type II Error () Test Conclusion H0 is False Type I Error () CORRECT 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 . IENG 486 Statistical Quality & Process Control

13. 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 1 0 0.10 0.01 H0not plausible Intermediate area H0plausible IENG 486 Statistical Quality & Process Control

14. 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 IENG 486 Statistical Quality & Process Control

15. 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 IENG 486 Statistical Quality & Process Control

16. 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 IENG 486 Statistical Quality & Process Control

17. 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 liquidAns: since the samples come from a normal distribution, and are added together in the process of computing the mean: IENG 486 Statistical Quality & Process Control

18. 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? IENG 486 Statistical Quality & Process Control

19. IENG 486 Statistical Quality & Process Control

20. 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? IENG 486 Statistical Quality & Process Control