The Power of Hypothesis Testing in Decision Making

Posted by on Dec 15, 2019 in Hypotheis Testing, Six Sigma | 0 comments

A statistical hypothesis is a statement or claim about some unrealized true state of nature.   It helps determine whether something is likely due to chance or to some other factor of interest.

 

Hypothesis testing is a powerful statistical technique that allows you to determine whether a certain factor or variable is statistically significant or not.

Hypothesis testing can help us make decisions.  Some examples might be:

  • Average gas mileage differs depending on whether the gasoline is purchased at Shell or BP.
  • The variability of machined thickness is dependent on the type of tool used.
  • Product quality is independent of the raw material supplier.
  • Whether the solution to our Six Sigma project actually made a difference or is the change we observe only due to chance.

The actual hypothesis tested consists of two statements about the true state of nature. H₀ is the null hypothesis and is the statement of a zero or null difference that is to be tested.  The other statement is the alternative hypothesis, H₁, and is the statement that must be true if the null hypothesis is false.

Other standard terms used in hypothesis testing are:

  • Type I error:  the mistake of rejecting the null hypothesis when it is true.
  • Type II error:  the mistake of failing to reject the null hypothesis when it is false
  • p-value: represents the exact probability of making a Type I error and is calculated from the data itself.

A common analogy of hypothesis testing can be taken from our legal system where an accused on trial is presupposed to be innocent unless the prosecution presents overwhelming evidence to convict him.  In this example, the hypotheses to be tested are stated as:

  • H₀:  Defendant is Innocent
  • H₁:  Defendant is guilty

Regardless of the jury’s conclusion, they are never really sure of the true state of nature.  Concluding “H₀: Defendant is Innocent” does not mean that the defendant is in fact innocent.  An H₀ conclusion simply means that the evidence was not overwhelming enough to justify a conviction.  On the other hand, concluding H₁ does not prove guilt; rather, it implies that the evidence is so overwhelming that the jury can have a high level of confidence in a guilty verdict.

Since verdicts are concluded with less than 100% certainty, either conclusion has some probability of error.  The probability of committing a Type I error is defined as alpha, α, and the probability of committing a Type II error is β.

In a courtroom, α, the probability of convicting an innocent person, is of critical concern.  To minimize the risk of such an erroneous conclusion, our courts require overwhelming evidence to conclude H₁.  Although minimizing α has its advantages, it should be obvious that requiring overwhelming evidence to conclude H₁ will in turn increase β, the probability of a Type II error.  To resolve this dilemma, hypothesis tests are designed such that:

  1. The most critical decision error is a Type I error.
  2. α is set at a minimum level, usually .05 or .01.
  3. Based on the above, the hypothesis statement to be tested for at least (1-α)100% confidence is placed in H₁.
  4. The nature of most statistical hypothesis tests require that the equality condition be placed in H₀.
  5. To minimize β while holding α constant requires increased sample sizes.

Let’s take another example and look at a situation where we want to reduce the variation in a process.  Our initial data is taken at the outset of our project.  We analyze our process, tighten it up by standardizing the process, control some important factors, implement our solutions and take another sample.

Initial data    Data with new solution
99.157           99.640
102.230         98.372
97.184            100.020
98.215           99.391
101.473          99.455
101.429         100.037
100.445         99.612
95.848          97.614
100.860         100.092
100.248         102.276
96.366          100.487
93.758           100.730
95.461           100.367
104.335         99.076
105.088         101.809
100.614          99.780
100.158          101.346
101.811           98.838
97.094           101.218
96.891           100.664
The mean and standard deviation of the two sets of data are:
                                                Mean      Standard Deviation
Initial data                             99.433             3.009
Data with new solution        100.04              1.14
Now the question may be asked since you only took 20 samples of each, how confident are you that the change you made in reducing the variation, expressed by the standard deviation, is real or did you get lucky with the sample you took?  An hypothesis test for two variances can be conducted on the samples to determine whether the difference observed is statistical significant. In this example our null hypothesis is the variance of the initial data sample is equal to the variance of the sample taken after the solution is implemented.  The alternative hypothesis is that they are not equal.  Using Minitab, we get the following results:

Test

Null hypothesis H₀: σ₁ / σ₂ = 1
Alternative hypothesis H₁: σ₁ / σ₂ ≠ 1
Significance level α = 0.05
Method Test
Statistic
DF1 DF2 P-Value
Bonett 13.17 1 0.000
Levene 12.06 1 38 0.001

 

From the above you can see that either method, Bonett or Levene, provides a p-value less than our significance level of α = 0.05.  Therefore we can conclude with 95% confidence that there is a statistical difference in the variance of the two samples.

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