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## Use quantitative reasoning to develop biological insights"Excellent site for both basic and advanced lessons on applying mathematics to biology."
- Tweeted by the Includes appendix of algebra, precalculus, and calculus. |

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[3] OA Liao and Tlsty,

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PDF Sample-variance curve fitting exercise for MatLab

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Even though academic tradition is a major contributor to the adoption of criteria for declaring statistical "significance," for example, the arbitrary *p* < 0.05 criterion, it is possible to choose criteria in a more rational manner. For example, a pharmaceutical company could compare potential research and legal costs with potential profits to conclude that it can only afford to move on to Phase III trials for experimental therapies that rule out a null hypothesis at a *p* < 0.01 level during Phase II trials.

95% confidence intervals do not necessarily thread the means of their associated parent distributions 95% of the time. Confidence intervals can be meaningless when derived from faulty assumptions.

Rejecting models having *p* < 0.1 does not ensure that 90% of surviving models are valid. Human imagination easily contrives a large number mutually incompatible models, each of which is consistent with a given set of experimental data at the *p* > 0.1 level. This consistency cannot logically guarantee that 90% of the models are valid since, by construction, most of them are false.

[7] PDF Goodman,

The ability of a system of DEs to produce solutions that resemble data does not rule out alternative descriptions of dynamics.

[8] PDF Ferrell, Jr., Tsai, and Yang,[10] OA Liao, Estévez-Salmerón, and Tlsty

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The success of a non-spatial model in mimicking aggregate population dynamics does not rule out a spatially-resolved model.

[15] PMC Macklin

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If closed universes (eventually) explore all accessible states equally, then small subsystems can be studied using partition functions

**DISCLAIMER:** The following video chapter is a simplified introduction that can be used in algebra-based high school physics classes. This is not a replacement for the references below.

[17] PDF Crooks

[18] OA England

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Calligraphy manifests the fundamental theorem of calculus:

- Thicker strokes cover paper at faster rates.
- Painted area depends both on length through which and on thickness(es) at which brush was drawn.