James Bond and six-sigma

Image by Thiru Murugan

In a Six Sigma project, once the required data has been collected, the next step is to make conclusions about certain parameters of the population using the sample data collected. To ensure that the right inferences have been drawn, the appropriate tests have been used, it is important to go back to understand the basics of hypothesis testing. I’ll do this using an example featuring someone called Bond, James Bond.

Null Hypothesis Testing:
In any statistical test of significance, we start with defining and identifying the null hypothesis H0 and the alternate hypothesis Ha. The next step is selecting the significance level or the ?. Then, we calculate the test statistic, which is the quantity calculated from our sample data. We compare this with the critical value, which is the value for the statistic calculated based on the significance level and decides whether or not to reject the null hypothesis in our test. If the observed value is greater than the critical value, we reject the null hypothesis and vice versa.

Setting up Hypotheses:
Setting up the hypotheses correctly is an essential part of statistical inference. Hypotheses are statements about parameters like expected value or variance for the population being studied or it could be a statement about the distributional form of the population. While formulating a hypothesis, usually some theory is put forward, because it is believed to be true and potentially impacts the argument for a change, but has not been proved earlier. This is also referred to as ‘Null Hypothesis’ or H0. The null hypothesis is accompanied by a competing claim against it called ‘Alternative Hypothesis’ or Ha, it is a statement of what a statistical hypothesis test is set up to establish.

The selection of the alternative hypothesis decides the nature of the test to use. Whether we are to calculate a two-tailed probability or a one tailed probability depends on nature of the hypothesis framed. Typical example for a two-tailed test would be when you only want to know if the means of two samples are different or not. The logic is simple, as the name suggests, in a two-tailed test we look at the rejection regions (refer the previous figure) on both the tails of the distribution, whereas in a one-tailed test we only look at the relevant side.

Usually the null hypothesis refers to status quo, and proposes that nothing special is happening. Then the alternate hypothesis would be that something special is happening, which what we test for. We will only accept the claim that the special thing exists if the experiment says that “nothing special happening” is very unlikely. That is, we accept the alternative hypothesis only if the probability of outcome of the null hypothesis is extremely low.

Vodka Martini, Shaken not Stirred …
James Bond always insisted that his Martinis should be shaken and not stirred. So we create an experiment to determine whether Mr. Bond could tell the difference between a shaken and a stirred martini. To do this we give Mr. Bond a series of 20 taste tests of stirred or shaken martinis and for each test, we flip a fair coin to determine whether to stir or shake the martini. Then we present the martini to Mr. Bond and ask him to decide whether it was shaken or stirred. Let’s say Mr. Bond was correct on 16 of the 20 taste tests. Does this prove that Mr. Bond has at least some ability to tell whether the martini was shaken or stirred? Could it be he was just lucky and guessed it right 16 out of 20 times. But how probable is the explanation that he was just lucky?

Using the binomial distribution calculator, we determine the probability for Mr. Bond to be guessing correctly 16/20 times or more is 0.0059 (calculated using a calculator on ). This is a pretty low probability, and therefore someone would have to be very lucky to be correct 16 or more times if he was just guessing. The point to note here is that the hypothesis that he was guessing is not proven false, only its likelihood to be true is very low. And so we can more confidently say that Mr. Bond can tell whether a martini is stirred or shaken.

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