11. Critical assumptions at the foundations of statistical analysis

Preamble: In previous activities, we have explored critical thinking about various modes of mathematical thinking—e.g., heuristic rules as alternatives to equations that predict trajectories; regression equations reflecting tightness of packing of points as an alternative to best prediction equationsor even causal relations. In this activity, we explore alternatives to the standard use of a t-test, which is a foundational method in statistical analysis for deciding whether something is happening.

Preparation
Taylor and Szteiter, "Statistical thinking," http://www.faculty.umb.edu/peter_taylor/StatisticalThinking.html

Goals
Explore the restrictiveness that comes with using any quantitative tools

Activity
1. Mini-lecture to review the t-test (based on reading above).
2. Workplace productivity activity
a. Each of you gets to represent 3 workers in a factory. For each worker use this applet, https://www.random.org/gaussian-distributions/, to generate your productivity. Write each on a separate card.

• (The distribution's mean should be 10 and its standard deviation 1.0 The numbers should have 2 3 significant digits. A result of, say, 1.8e+1 means 18, i.e., 1.8 *10)

b. The factory owner thinks that piping in music will increase productivity of the workers. For each of your workers, generate the new productivity using the applet. For each worker the distribution's mean should be changed to the original productivity for that worker (as written on the card) plus .5. (The standard deviation remains the same.) Write this below the other figure.
c. Input both numbers for each worker into the google form, http://bit.ly/2wgBfDL .
d. Take note of the "P-value" for the output, http://bit.ly/2fazL8z . This value expresses how likely it is that the difference between the average before and after could happen if the factory productivity had not been influenced by the music. Should the employer pipe in music?
e. If you have a worker whose productivity went down, invent an explanation why. Bring this into the discussion, which the enlightened employer will convene to get ideas about other paths to higher overall productivity that the employer could foster.
f. Suppose now that the enlightened employer goes a step further and shows interest in the dynamics (or mechanisms or causal connections) through which factors in addition to music influence productivity. Invent something that each of your workers might say and bring that into the discussion.
3. Mini-lecture (time permitting; based on Taylor 2013) to a) contrast the standard use of a t-test—are averages significantly different?—to the more participatory modes (in #e & f) for exposing a variety of responses underneath observed variation; and b) stimulate critical thinking about claims concerning differences in average test scores across racially defined groups in the USA. visual aids
4. Plus-delta feedback on activity
5. Post-class reading (optional): Taylor, "Who can act?..." which reviews the steps in the activity.

References
Taylor, P.J. (2010) "Who can act? Critical assumptions at the foundations of statistical analysis: Explaining differences among means – What can that mean?" manuscript pdf
Taylor, P. J. (2013) "A lesson in race, genes, and IQ," GeneWatch 26 (2-3): 33-35, http://www.councilforresponsiblegenetics.org/GeneWatch/GeneWatchPage.aspx?pageId=484 (viewed 10 Sep 17)