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Say I am tracking my sleep for 1 week and these are the number of hours I sleep each night:

(5, 6, 9, 4, 8, 9, 6)

everyday that I track my sleep I am also taking a test that measures how well my brain is working these were my scores:

(45, 23, 33, 48, 68, 19, 26)

The higher the score the better.

Given these numbers, I would like to find out the optimal number of hours I would have to sleep to score the highest on this test. For example, the answer could be X amount of hours would give me the best chance of scoring highest on this test. I would like to know how to find X.

Any help that points me in the right direction would be incredibly appreciated!

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Now, I am assuming you just pulled those numbers out of a hat, because when I plotted them, I saw absolutely NO relationship between them.

Theoretically, you will first need to have an idea of the underlying model, or a reasonably accurate approximation. For example, a quadratic model for the mean test score might be useful since it can accommodate linear (no limits to the benefit of sleep) and non-linear (diminishing or negative effects of too much sleep).

Then you will need to determine the form of the variability. I doubt normality will hold, so you may want to examine the residuals for a least squares fit to give you some ideas, then use a generalized linear model with the proper mean-variance relationship and a strictly positive range.

The alternative is to use a non-parametric regression...which is a vast field of techniques.

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  • $\begingroup$ Thanks for this answer, I am going to try to apply it and I will get back to you. $\endgroup$ – ahat91 Oct 3 '14 at 14:09
  • $\begingroup$ I am having difficulty, I realize this is a lot to ask for but is there anyway you can show me an actual example using similar numbers to the ones above? I am creating a mobile application and finding the answer to this question is a cornerstone to my app. This would be greatly appreciated! I'd love to talk with you more if you have any further questions. $\endgroup$ – ahat91 Oct 3 '14 at 14:43

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