Analyze Phase Introduction to Hypothesis Testing

1. Analyze PhaseIntroduction to Hypothesis Testing

2. Hypothesis Testing (ND) Welcome to Analyze “X” Sifting Inferential Statistics Hypothesis Testing Purpose 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

3. 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.

4. 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) Purpose of Hypothesis Testing

5. 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. The Basic Concept for Hypothesis Tests Did my sample come from this population? Or this? Or this?

6. 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? Significant Difference   Sample 2 Sample 1

7. 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, practicaland statistical. Failure to tie these two differences together is one of the most common errors in statistics. Detecting Significance HO: The sky is not falling. HA: The sky is falling.

8. 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. Practical vs. Statistical We will see that it is possible to realize a statistically significant difference without realizing a practically significant difference.

9. During the Measure Phase, it is important that the nature of the problem be well understood. 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. Detection of a difference is then accomplished using statistical Hypothesis Testing. Detecting Significance Mean Shift Variation Reduction

10. 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. Hypothesis Testing

11. 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. DICE Example

12. Hypothesis Testing Type I Error α DECISIONS Sample Size n β Type II Error

13. 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. Statistical Hypotheses P-value > 0.05 Ho = no difference or relationship P-value < 0.05 Ha = is a difference or relationship Making a decision does not FIX a problem, taking action does.

14. State the Practical Problem. State the Statistical Problem. HO: ___ = ___ HA: ___ ≠ ,>,< ___ Select the appropriate statistical test and risk levels. α = .05 β = .10 Establish the sample size required to detect the difference. State the Statistical Solution. State the Practical Solution. Steps to Statistical Hypothesis Test Noooot THAT practical solution!

15. 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%. How Likely is Unlikely?

16. 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 Different (Ho is True) (Ho is False) Correct Decision Type II Error Not Different (Fail to Reject Ho) StatisticalConclusions Correct Decision Type 1 Error Different (Reject Ho) Hypothesis Testing Risk

17. Alpha ( ) risks are expressed relative to a reference distribution. Distributions include: t-distribution z-distribution 2- distribution F-distribution Accept as chance differences Region of DOUBT Region of DOUBT Alpha Risk The a-level is represented by the clouded areas. Sample results in this area lead to rejection of H0.

18. 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 Different (Ho is True) (Ho is False) Correct Decision Type II Error Not Different (Fail to Reject Ho) StatisticalConclusions Correct Decision Type 1 Error Different (Reject Ho) Hypothesis Testing Risk

19. Beta Risk is the probability of failing to reject the null hypothesis when a difference exists. Reject H0  = Pr(Type 1 error) Beta Risk Distribution if H0 is true  = 0.05 H0 value Accept H0 Distribution if Ha is true = Pr(Type II error)  Critical value of test statistic

20. Distinguishing between Two Samples Theoretical Distribution of Means When n = 2 d = 5 S = 1  Recall from the Central Limit Theorem as the number of individual observations increase the Standard Error decreases. 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. Theoretical Distribution of Means When n = 30 d = 5 S = 1

21. Large Delta  Large S 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. 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. We want to be 95% confident in all of our estimates!

22. Question: “How many samples should we take?” Answer: “Well, that depends on the size of your delta and Standard Deviation”. Question: “How should we conduct the sampling?”Answer: “Well, that depends on what you want to know”. Question: “Was the sample we took large enough?”Answer: “Well, that depends on the size of your delta and Standard Deviation”. Question: “Should we take some more samples just to be sure?”Answer: “No, not if you took the correct number of samples the first time!” Typical Questions on Sampling

23. 50 Population 40 60 70 40 40 50 50 60 60 70 70 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 Non-normal Data, multiply your calculated sample size by 1.1

24. Normal Continuous Data Test of Equal Variance 1 Sample Variance 1 Sample t-test Variance Equal Variance Not Equal 2 Sample T One Way ANOVA 2 Sample T One Way ANOVA Hypothesis Testing Roadmap

25. Continuous Data Non Normal Test of Equal Variance Median Test Mann-Whitney Several Median Tests Hypothesis Testing Roadmap

26. Attribute Data Attribute Data One Factor Two Factors Two Samples Two or More Samples One Sample One Sample Proportion Two Sample Proportion Chi Square Test (Contingency Table) Minitab: Stat - Basic Stats - 2 Proportions If P-value < 0.05 the proportions are different 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 Hypothesis Testing Roadmap

27. 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.” Common Pitfalls to Avoid

28. 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 Summary