Random Digits, Simulation and Chance Differences

Using a table of random digits

To choose an SRS, or to assign treatments, the exam expects a procedure that anyone could repeat: (1) label every unit with the same number of digits — $40$ units get labels $01$ to $40$; (2) read the table in groups of that many digits; (3) ignore numbers outside the label range and any repeats; (4) stop when you have enough. With the line $71340\ 08521\ 45663$ and labels $01$–$40$, the pairs $71,\,34,\,00,\,85,\,21,\,45,\,66,\,3\ldots$ give the units $34$ and $21$ first. For random assignment, the first $20$ labels drawn go to treatment A and the rest to B — write that sentence, it is the part graders look for.

Is the difference real or just chance?

After an experiment you see, say, a mean difference of $8$ points between the two groups. Random assignment alone would produce some difference even if the treatment did nothing. A simulation answers "how big a difference does chance produce?": keep the observed responses, re-randomise the group labels many times, compute the difference each time, and count how often it is at least as large as the one observed. If only $6$ of $200$ re-randomisations reach $8$ points, chance produces such a gap about $3\%$ of the time — the result is statistically significant, meaning unlikely to be due to random assignment alone.

Describing a simulation

A full-credit description names four things: what one trial represents, how randomness is generated (digits, cards, a spinner), what is recorded from each trial, and how the many trials are used to estimate the probability. Missing any one of the four costs the point.