Analyze Phase Introduction to Hypothesis Testing 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 P 1 ANOVA Hypothesis Testing ND P 2 Hypothesis Testing NND P 1 Hypothesis Testing NND P 2 Wrap Up & Action Items OSSS LSS Black Belt v 9. 1 - 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 whethere are different population parameters or that the differences are just due to expected sample variation. OSSS LSS Black Belt v 9. 1 - Analyze Phase 3 © 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 v 9. 1 - 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. Did my sample come from this population? Or this? OSSS LSS Black Belt v 9. 1 - 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 v 9. 1 - 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 v 9. 1 - 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 v 9. 1 - 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 v 9. 1 - 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 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 v 9. 1 - 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 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 v 9. 1 - Analyze Phase 11 © Open. Source. Six. Sigma, LLC

Hypothesis Testing α DECISIONS n β OSSS LSS Black Belt v 9. 1 - Analyze Phase 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 v 9. 1 - 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 v 9. 1 - Analyze Phase 14 © Open. Source. Six. Sigma, LLC

How Likely is Unlikely? Any differences between observed data and claims made under H 0 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 H 0 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 v 9. 1 - 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 v 9. 1 - 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 H 0. distribution – F-distribution Region of DOUBT Accept as chance differences OSSS LSS Black Belt v 9. 1 - 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 Different (Reject Ho) OSSS LSS Black Belt v 9. 1 - Analyze Phase Different (Ho is False) Correct Decision Type II Error Type 1 Error Correct Decision 18 © Open. Source. Six. Sigma, LLC

Beta Risk is the probability of failing to reject the null hypothesis when a difference exists. Distribution if H 0 is true Reject H 0 = Pr(Type 1 error) = 0. 05 H 0 value Accept H 0 = Pr(Type II error) Distribution if Ha is true Critical value of test statistic OSSS LSS Black Belt v 9. 1 - 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. Theoretical Distribution of Means When n = 2 =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 v 9. 1 - Analyze Phase 20 Theoretical Distribution of Means When n = 30 =5 S=1 © Open. Source. Six. Sigma, LLC

Delta Sigma—The Ratio between d and S Delta ( ) 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 When � & 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 v 9. 1 - 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 v 9. 1 - 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 Population 40 OSSS LSS Black Belt v 9. 1 - Analyze Phase 50 23 50 © Open. Source. Six. Sigma, LLC

Hypothesis Testing Roadmap Normal us o nu i t n Co Data Test of Equal Variance 1 Sample Variance Equal 2 Sample T 1 Sample t-test Variance Not Equal One Way ANOVA OSSS LSS Black Belt v 9. 1 - Analyze Phase 2 Sample T 24 One Way ANOVA © Open. Source. Six. Sigma, LLC

Hypothesis Testing Roadmap us o nu i t n Co Data Non Normal Test of Equal Variance Mann-Whitney OSSS LSS Black Belt v 9. 1 - Analyze Phase Median Test Several Median Tests 25 © Open. Source. Six. Sigma, LLC

Hypothesis Testing Roadmap Attribute Data te u b ri Att ata D One Factor Two Samples 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 v 9. 1 - 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 v 9. 1 - 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 v 9. 1 - Analyze Phase 28 © Open. Source. Six. Sigma, LLC