Are you telepathic or telepathetic?

3.1 The psychic experiment

Working in pairs, designate one person as the sender and one person as the receiver. (We’ll swap roles in a moment.)

Here’s the procedure for each attempted transmission. The sender thinks of a number between one to five, attempting to transmit it to the receiver by the power of thought alone (no using thoughts to activate muscles!). Within five seconds, the receiver makes a guess about the number. The sender then indicates whether it is correct, and the receiver writes down the outcome.

Repeat the procedure above 20 times. Then swap roles and work through another 20 attempts.

Once you have your data, calculate the proportion of correct responses. What was the expected proportion? Are your results weird? (Are you psychic? Or anti-psychic?)

If you finish early, try a replication. Do you get the same results?

3.1 The psychic experiment

What was the expected proportion of correct responses? 1 in 5 = 0.20

To answer the second question (are your–or anyone’s–results weird?), let’s turn to simulation.

3.2 The simulation

  • Create a simulation of the experiment.

  • Then run it 10,000 times by using a for loop.

  • Finally, plot the results.

3.2 The simulation

proportion_correct <- numeric(10000) 
for (i in 1:10000) {
  simulated_attempts <- sample(c('Correct', 'Incorrect'), 
                               size = 20, prob = c(0.2, 0.8), replace = TRUE)
  proportion_correct[i] <- mean(simulated_attempts == 'Correct')
}
hist(proportion_correct, freq=FALSE, 
     breaks=seq(-0.025,max(proportion_correct)+0.025, 0.05))

3.3 Reasoning by using the distribution

How weird were the proportions you observed? How often would you expect to get the score you found just by chance? How often would you expect to find that score or an even higher one?

Do you think that chance fully accounts for your score?

If you got a rather high score, what should you do next?

3.3 Reasoning by using the distribution

How often would you expect to get the score you found just by chance? How often would you expect to find that score or an even higher one?

Consider a proportion of 0.4. How often would you expect to score 8 of 20?

mean(proportion_correct == 0.4)
## [1] 0.0225

What about scoring at least an 8 of 20?

mean(proportion_correct >= 0.4)
## [1] 0.0327

An 8 or an 8+ are fairly rare, but we’ll probably see them today.

3.3 Reasoning by using the distribution

What about the anti-psychics? Is 0 out of 20 uncommon?

mean(proportion_correct == 0)
## [1] 0.0133

A 0 is rare, but not outlandishly so.

3.3 Reasoning by using the distribution

Do you think that chance fully accounts for your score?

Probably not. People aren’t good random number generators, so the sender probably had fewer repeats (e.g. 1, then 1 again) than a random sequence would. People also think that if a number hasn’t come up in a while, it is “due”. Finally, the sender may have communicated non-telepathically, whether deliberately or not. That’s a potential confound.

If you got a rather high score, what should you do next?

  • Replicate.

  • Look for confounds.

  • Profit. (Okay, don’t do this. But people who do emphasize their rare–but expected–successes and ignore their failures!)

Why did we do this activity?

Let’s make the goals explicit.

  • Our primary aim was not to discover whether anyone was telepathic or not. Your instructors are fairly confident that the answer is “no”, and today’s test wasn’t a particularly good one for assessing telepathy per se.

  • Instead, we wanted you to explore a distribution of outcomes, getting a feel for how likely different outcomes are through simulation. Rare events may still be expected with enough repetitions.

  • We also wanted you to consider the non-chance elements that influence the observed numbers. These include different types of bias (non-random sequence generation, confounds) in the experiment.