#### Transcript Analyze - Intro to Hypothesis Testing

Analyze Phase Introduction to Hypothesis Testing Hypothesis Testing (ND) Welcome to Analyze “X” Sifting Hypothesis Testing Purpose Inferential Statistics Tests for Central Tendency Intro to Hypothesis Testing Tests for Variance Hypothesis Testing ND P1 ANOVA Hypothesis Testing ND P2 Hypothesis Testing NND P1 Hypothesis Testing NND P2 Wrap Up & Action Items OSSS LSS Black Belt v10.0 - Analyze Phase 2 © Open Source Six Sigma, LLC Six Sigma Goals and Hypothesis Testing Our goal is to improve our Process Capability, this translates to the need to move the process Mean (or proportion) and reduce the Standard Deviation. – Because it is too expensive or too impractical (not to mention theoretically impossible) to collect population data, we will make decisions based on sample data. – Because we are dealing with sample data, there is some uncertainty about the true population parameters. Hypothesis Testing helps us make fact-based decisions about whether there are different population parameters or that the differences are just due to expected sample variation. Process Capability of Process Before LSL Process Capability of Process After USL P rocess Data LS L 100.00000 Target * USL 120.00000 S ample M ean 108.65832 S ample N 150 S tD ev (Within) 2.35158 S tD ev (O v erall) 5.41996 LSL Within Ov erall P otential (Within) C apability Cp 1.42 C PL 1.23 C PU 1.61 C pk 1.23 C C pk 1.42 USL P rocess Data LS L 100.00000 Target * USL 120.00000 S ample M ean 109.86078 S ample N 100 S tD ev (Within) 1.55861 S tD ev (O v erall) 1.54407 Within Ov erall P otential (Within) C apability Cp 2.14 C PL 2.11 C PU 2.17 C pk 2.11 C C pk 2.14 O v erall C apability Pp PPL PPU P pk C pm 96 O bserv ed P erformance P P M < LS L 6666.67 PPM > USL 0.00 P P M Total 6666.67 100 E xp. Within P erformance P P M < LS L 115.74 PPM > USL 0.71 P P M Total 116.45 104 108 112 116 Pp PPL PPU P pk C pm 120 102 E xp. O v erall P erformance P P M < LS L 55078.48 P P M > U S L 18193.49 P P M Total 73271.97 OSSS LSS Black Belt v10.0 - Analyze Phase O v erall C apability 0.62 0.53 0.70 0.53 * O bserv ed P erformance P P M < LS L 0.00 P P M > U S L 0.00 P P M Total 0.00 3 105 E xp. Within P erformance P P M < LS L 0.00 P P M > U S L 0.00 P P M Total 0.00 108 111 114 117 2.16 2.13 2.19 2.13 * 120 E xp. O v erall P erformance P P M < LS L 0.00 P P M > U S L 0.00 P P M Total 0.00 © Open Source Six Sigma, LLC Purpose of Hypothesis Testing The purpose of appropriate Hypothesis Testing is to integrate the Voice of the Process with the Voice of the Business to make data-based decisions to resolve problems. Hypothesis Testing can help avoid high costs of experimental efforts by using existing data. This can be likened to: – Local store costs versus mini bar expenses. – There may be a need to eventually use experimentation, but careful data analysis can indicate a direction for experimentation if necessary. The probability of occurrence is based on a pre-determined statistical confidence. Decisions are based on: – Beliefs (past experience) – Preferences (current needs) – Evidence (statistical data) – Risk (acceptable level of failure) OSSS LSS Black Belt v10.0 - Analyze Phase 4 © Open Source Six Sigma, LLC The Basic Concept for Hypothesis Tests Recall from the discussion on classes and cause of distributions that a data set may seem Normal, yet still be made up of multiple distributions. Hypothesis Testing can help establish a statistical difference between factors from different distributions. 0.8 0.7 0.6 freq 0.5 0.4 0.3 0.2 0.1 0.0 -3 -2 -1 0 1 2 3 x Did my sample come from this population? Or this? Or this? OSSS LSS Black Belt v10.0 - Analyze Phase 5 © Open Source Six Sigma, LLC Significant Difference Are the two distributions “significantly” different from each other? How sure are we of our decision? How do the number of observations affect our confidence in detecting population Mean? Sample 2 Sample 1 OSSS LSS Black Belt v10.0 - Analyze Phase 6 © Open Source Six Sigma, LLC Detecting Significance Statistics provide a methodology to detect differences. – Examples might include differences in suppliers, shifts or equipment. – Two types of significant differences occur and must be well understood, practical and statistical. – Failure to tie these two differences together is one of the most common errors in statistics. HO: The sky is not falling. HA: The sky is falling. OSSS LSS Black Belt v10.0 - Analyze Phase 7 © Open Source Six Sigma, LLC Practical vs. Statistical Practical Difference: The difference which results in an improvement of practical or economic value to the company. – Example, an improvement in yield from 96 to 99 percent. Statistical Difference: A difference or change to the process that probably (with some defined degree of confidence) did not happen by chance. – Examples might include differences in suppliers, markets or servers. We will see that it is possible to realize a statistically significant difference without realizing a practically significant difference. OSSS LSS Black Belt v10.0 - Analyze Phase 8 © Open Source Six Sigma, LLC Detecting Significance During the Measure Phase, it is important that the nature of the problem be well understood. Mean Shift In understanding the problem, the practical difference to be achieved must match the statistical difference. The difference can be either a change in the Mean or in the variance. Variation Reduction Detection of a difference is then accomplished using statistical Hypothesis Testing. OSSS LSS Black Belt v10.0 - Analyze Phase 9 © Open Source Six Sigma, LLC Hypothesis Testing A Hypothesis Test is an a priori theory relating to differences between variables. A statistical test or Hypothesis Test is performed to prove or disprove the theory. A Hypothesis Test converts the practical problem into a statistical problem. – Since relatively small sample sizes are used to estimate population parameters, there is always a chance of collecting a non-representative sample. – Inferential statistics allows us to estimate the probability of getting a non-representative sample. OSSS LSS Black Belt v10.0 - Analyze Phase 10 © Open Source Six Sigma, LLC DICE Example We could throw it a number of times and track how many each face occurred. With a standard die, we would expect each face to occur 1/6 or 16.67% of the time. If we threw the die 5 times and got 5 one’s, what would you conclude? How sure can you be? – Pr (1 one) = 0.1667 Pr (5 ones) = (0.1667)5 = 0.00013 There are approximately 1.3 chances out of 1000 that we could have gotten 5 ones with a standard die. Therefore, we would say we are willing to take a 0.1% chance of being wrong about our hypothesis that the die was “loaded” since the results do not come close to our predicted outcome. OSSS LSS Black Belt v10.0 - Analyze Phase 11 © Open Source Six Sigma, LLC Hypothesis Testing α DECISIONS β OSSS LSS Black Belt v10.0 - Analyze Phase n 12 © Open Source Six Sigma, LLC Statistical Hypotheses A hypothesis is a predetermined theory about the nature of, or relationships between variables. Statistical tests can prove (with a certain degree of confidence) that a relationship exists. We have two alternatives for hypothesis. – The “null hypothesis” Ho assumes that there are no differences or relationships. This is the default assumption of all statistical tests. – The “alternative hypothesis” Ha states that there is a difference or relationship. P-value > 0.05 P-value < 0.05 Ho = no difference or relationship Ha = is a difference or relationship Making a decision does not FIX a problem, taking action does. OSSS LSS Black Belt v10.0 - Analyze Phase 13 © Open Source Six Sigma, LLC Steps to Statistical Hypothesis Test 1. State the Practical Problem. 2. State the Statistical Problem. a) HO: ___ = ___ b) HA: ___ ≠ ,>,< ___ 3. Select the appropriate statistical test and risk levels. a) α = .05 b) β = .10 4. Establish the sample size required to detect the difference. 5. State the Statistical Solution. 6. State the Practical Solution. Noooot THAT practical solution! OSSS LSS Black Belt v10.0 - Analyze Phase 14 © Open Source Six Sigma, LLC How Likely is Unlikely? Any differences between observed data and claims made under H0 may be real or due to chance. Hypothesis Tests determine the probabilities of these differences occurring solely due to chance and call them P-values. The a level of a test (level of significance) represents the yardstick against which P-values are measured and H0 is rejected if the P-value is less than the alpha level. The most commonly used levels are 5%, 10% and 1%. OSSS LSS Black Belt v10.0 - Analyze Phase 15 © Open Source Six Sigma, LLC Hypothesis Testing Risk The alpha risk or Type 1 Error (generally called the “Producer’s Risk”) is the probability that we could be wrong in saying that something is “different.” It is an assessment of the likelihood that the observed difference could have occurred by random chance. Alpha is the primary decision-making tool of most statistical tests. Actual Conditions Not Different (Ho is True) Not Different (Fail to Reject Ho) Statistical Conclusions (Ho is False) Correct Decision Type II Error Type 1 Error Correct Decision Different (Reject Ho) OSSS LSS Black Belt v10.0 - Analyze Phase Different 16 © Open Source Six Sigma, LLC Alpha Risk Alpha ( ) risks are expressed relative to a reference distribution. Distributions include: – t-distribution The a-level is represented by the clouded areas. – z-distribution – 2- Sample results in this area lead to rejection of H0. distribution – F-distribution Region of DOUBT Region of DOUBT Accept as chance differences OSSS LSS Black Belt v10.0 - Analyze Phase 17 © Open Source Six Sigma, LLC Hypothesis Testing Risk The beta risk or Type 2 Error (also called the “Consumer’s Risk”) is the probability that we could be wrong in saying that two or more things are the same when, in fact, they are different. Actual Conditions Not Different (Ho is True) Not Different (Fail to Reject Ho) Statistical Conclusions Correct Decision Type II Error Type 1 Error Correct Decision Different (Reject Ho) OSSS LSS Black Belt v10.0 - Analyze Phase Different (Ho is False) 18 © Open Source Six Sigma, LLC Beta Risk Beta Risk is the probability of failing to reject the null hypothesis when a difference exists. Distribution if H0 is true Reject H0 = Pr(Type 1 error) = 0.05 H0 value Accept H0 = Pr(Type II error) Distribution if Ha is true Critical value of test statistic OSSS LSS Black Belt v10.0 - Analyze Phase 19 © Open Source Six Sigma, LLC Distinguishing between Two Samples Recall from the Central Limit Theorem as the number of individual observations increase the Standard Error decreases. d Theoretical Distribution of Means When n = 2 d=5 S=1 In this example when n=2 we cannot distinguish the difference between the Means (> 5% overlap, P-value > 0.05). When n=30, we can distinguish between the Means (< 5% overlap, P-value < 0.05) There is a significant difference. OSSS LSS Black Belt v10.0 - Analyze Phase 20 Theoretical Distribution of Means When n = 30 d=5 S=1 © Open Source Six Sigma, LLC Delta Sigma—The Ratio between d and S Delta (d) is the size of the difference between two Means or one Mean and a target value. Sigma (S) is the sample Standard Deviation of the distribution of individuals of one or both of the samples under question. Large Delta d When d & S is large, we don’t need statistics because the differences are so large. If the variance of the data is large, it is difficult to establish differences. We need larger sample sizes to reduce uncertainty. Large S We want to be 95% confident in all of our estimates! OSSS LSS Black Belt v10.0 - Analyze Phase 21 © Open Source Six Sigma, LLC Typical Questions on Sampling Question: “How many samples should we take?” Answer: “Well, that depends on the size of your delta and Standard Deviation”. Question: Answer: “How should we conduct the sampling?” “Well, that depends on what you want to know”. Question: Answer: “Was the sample we took large enough?” “Well, that depends on the size of your delta and Standard Deviation”. Question: Answer: “Should we take some more samples just to be sure?” “No, not if you took the correct number of samples the first time!” OSSS LSS Black Belt v10.0 - Analyze Phase 22 © Open Source Six Sigma, LLC The Perfect Sample Size The minimum sample size required to provide exactly 5% overlap (risk). In order to distinguish the Delta. Note: If you are working with Nonnormal Data, multiply your calculated sample size by 1.1 40 60 70 60 70 Population 40 OSSS LSS Black Belt v10.0 - Analyze Phase 50 23 50 © Open Source Six Sigma, LLC Hypothesis Testing Roadmap Normal Test of Equal Variance 1 Sample Variance Variance Equal 2 Sample T 1 Sample t-test Variance Not Equal One Way ANOVA OSSS LSS Black Belt v10.0 - Analyze Phase 2 Sample T 24 One Way ANOVA © Open Source Six Sigma, LLC Hypothesis Testing Roadmap Non Normal Test of Equal Variance Mann-Whitney OSSS LSS Black Belt v10.0 - Analyze Phase Median Test Several Median Tests 25 © Open Source Six Sigma, LLC Hypothesis Testing Roadmap Attribute Data One Factor Two Samples One Sample One Sample Proportion Two Sample Proportion Minitab: Stat - Basic Stats - 2 Proportions If P-value < 0.05 the proportions are different Two Factors Two or More Samples Chi Square Test (Contingency Table) Minitab: Stat - Tables - Chi-Square Test If P-value < 0.05 at least one proportion is different Chi Square Test (Contingency Table) Minitab: Stat - Tables - Chi-Square Test If P-value < 0.05 the factors are not independent OSSS LSS Black Belt v10.0 - Analyze Phase 26 © Open Source Six Sigma, LLC Common Pitfalls to Avoid While using Hypothesis Testing the following facts should be borne in mind at the conclusion stage: – – – – The decision is about Ho and NOT Ha. The conclusion statement is whether the contention of Ha was upheld. The null hypothesis (Ho) is on trial. When a decision has been made: • Nothing has been proved. • It is just a decision. • All decisions can lead to errors (Types I and II). – If the decision is to “Reject Ho,” then the conclusion should read “There is sufficient evidence at the α level of significance to show that “state the alternative hypothesis Ha.” – If the decision is to “Fail to Reject Ho,” then the conclusion should read “There isn’t sufficient evidence at the α level of significance to show that “state the alternative hypothesis.” OSSS LSS Black Belt v10.0 - Analyze Phase 27 © Open Source Six Sigma, LLC Summary At this point, you should be able to: • Articulate the purpose of Hypothesis Testing • Explain the concepts of the Central Tendency • Be familiar with the types of Hypothesis Tests OSSS LSS Black Belt v10.0 - Analyze Phase 28 © Open Source Six Sigma, LLC