Reading the chi-square table

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manpreet Tuteehub forum best answer Best Answer 2 years ago

 

For solving problems involving the chi-square distribution in my syllabus, we use this χ2χ2 statistical table. However, I have a doubt in reading it.

Suppose there is an experiment where a die is thrown many times. We tabulate the times for which 11turns up, 22 turns up and so on. Now we want to check whether the die is fair.

We can construct an expected values table, in which number of times each digit must show up will be 1/61/6 times the total number of throws. Then we calculate χ2=(OiEi)2Eiχ2=∑(Oi−Ei)2Ei where OiOi and EiEiare the observed and the expected values at the ithith trial respectively.

Then we consult the table. Degrees of freedom will be ν=61=5ν=6−1=5. Now, my text says that the critical point of rejection is p=0.05p=0.05. That means, if 

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manpreet 2 years ago

There is no such thing as "95% fair".

The fact is you cannot state with absolute certainty that the die is unbiased. The best you can do is say: if it's fair then it should give approximately-equal counts (low χ2χ2 value); so if the χ2χ2 value is too high then, with some quantified risk (in your case 5%) of being wrong, we will reject the hypothesis of fairness.

So it's a trade-off. You have to decide how to balance the risk of accepting a loaded die versus the risk of rejecting a fair one. In the real world, that could come down profit, reputation, whatever. Set the χ2χ2 threshold too low and you'll reject too many fair dice; set it too high and you'll accept too many loaded ones.

If you do more trials (more rolls of the die), you'll get a cleaner separation between the behaviour of fair and loaded dice, and lower risk of a wrong decision. But again in the real world there would be a practical issue: is the improved ability to distinguish fair from loaded really worth the cost of rolling the die another however-many times?


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