Questions from an imaginary statistical methods exam

by RichardKennaway 5y4th Feb 201525 comments

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Answers to these questions should be expressed numerically, where possible, but no number should be given without a justification for the specific value.

1. Suppose that you have mislaid your house keys, something most people have experienced at one time or another. You look in various places for them: where you remember having them last, places you've been recently, places they should be, places they shouldn't be, places they couldn't be, places you've looked already, and so on. Eventually, you find them and stop looking.

Every time you looked somewhere, you were testing a hypothesis about their location. You may have looked in a hundred places before finding them.

As a piece of scientific research to answer the question "where are my keys?", this procedure has massive methodological flaws. You tested a hundred hypotheses before finding one that the data supported, ignoring every failed hypothesis. You really wanted each of these hypotheses in turn to be true, and made no attempt to avoid bias. You stopped collecting data the moment a hypothesis was confirmed. When you were running out of ideas to test, you frantically thought up some more. You repeated some failed experiments in the hope of getting a different result. Multiple hypotheses, file drawer effect, motivated cognition, motivated stopping, researcher degrees of freedom, remining of old data: there is hardly a methodological sin you have not committed.

(a) Should these considerations modify your confidence or anyone else's that you have in fact found your keys? If not, why not, and if so, what correction is required?

(b) Should these considerations affect your subsequent decisions (e.g. to go out, locking the door behind you)?

2. You have a lottery ticket. (Of course, you are far too sensible to ever buy such a thing, but nevertheless suppose that you have one. Maybe it was an unexpected free gift with your groceries.) The lottery is to be drawn later that day, the results available from a web site whose brief URL is printed on the ticket. You calculate a chance of about 1 in 100 million of a prize worth getting excited about.

(a) Once the lottery results are out, do you check your ticket? Why, or why not?

(b) Suppose that you do, and it appears that you have won a very large sum of money. But you remember that the prior chance of this happening was 1 in 100 million. How confident are you at this point that you have won? What alternative hypotheses are also raised to your attention by the experience of observing the coincidence of the numbers on your ticket and the numbers on the lottery web site?

(c) Suppose that you go through the steps of contacting the lottery organisers to make a claim, having them verify the ticket, collecting the prize, seeing your own bank confirm the deposit, and using the money in whatever way you think best. At what point, if any, do you become confident that you really did win the lottery? If never, what alternative hypotheses are you still seriously entertaining, to the extent of acting differently on account of them?

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