Psychic Probability | Case Closed

AP STATISTICS | Psychic Probability | Case Closed!    

Psychic Probability Researchers in parapsychology must be both wary of tricksters and able to discount instances of luck when investigating evidence of psychic ability. This is especially true in the testing of individuals who claim to possess these extraordinary abilities. In one such test of psychic ability, researchers claim to have eliminated the possibility of trickery by a self-proclaimed psychic and report that his ability exceeded that dictated by chance. The subject of this study was the controversial psychic Olof Jonsson. The test was conducted using a computer, a video-game paddle, an internal number generator, and a modified version of a computer program known as the ESPerciser. For each trial, the ESPerciser program began with a screen with four blank rectangles and the word “IMPRESSION PERIOD – PRESS BUTTON WHEN READY.” After the subject clicked the paddle button, the program would display four black-and-white symbols (star, wave, cross, and circle) in a random order on the screen. This was followed by a screen with all four symbols and the words “USE PADDLE TO POINT, PRESS BUTTON TO SELECT.” The subject was supposed to select the symbol he thought the computer had picked. After the subject had selected one of the symbols, the program asked the subject to indicate his level of confidence in his selection as low, medium, or high. If the subject’s guess matched the symbol chosen by the computer, the word “HIT” was repeatedly flashed in large multicolored letters, along with sound effects. The experiment was conducted in three sessions on three consecutive days. Each session consisted of four runs of 24 trials for an experiment total of 288 trials. For each run, there was 50% chance that the computer was in precognition mode, in which the computer selected the symbol after the subject made a guess, and there was a 50% chance that it was in a clairvoyance mode, in which the computer made its selection at the start of each trial. The subject was not informed of the mode until after the run was completed. We want to determine if the subject, Olof Jonsson, obtained results significantly better than chance in this experiment. The results are presented in the following table: Data Breakdown All trials High confidence Medium confidence Low confidence Precognition mode Clairvoyance mode

Trials 288 165 48 75 72 216

Hits 88 55 12 21 27 61

a) Compute the proportion of success for all trials and separately for the high-confidence, mediumconfidence, and low-confidence trials using the data from the table. If we assume that the trials are independent and the subject is simply guessing, what would the expected proportion of successes be in each case? Comment on the observed versus expected proportions. b) If the computer selected one of the four black-and-white symbols independently and at random on each trial, is the overall result of 88 hits in 288 trials evidence of psychic ability? Use an appropriate statistical procedure to justify your answer. c) The paper by the researchers does not indicate whether the trials were independent of whether the ESPerciser functions like a deck of 24 cards with trials like draws from the deck without

AP STATISTICS | Psychic Probability | Case Closed!    

replacement. In (a) we assumed that the trials were independent (draws from a deck of cards with replacement). We now investigate what the effect might be if trials were like draws of cards without replacement. To keep things simple, assume that we have a deck with only four cards, one of each symbol. What is the probability of making a correct guess on the first, second, third, and fourth draws if you are told what happens on each draw (that is, after each guess, you are shown the actual card)? d) Under the same assumptions as in (b), if you are just guessing, how many guesses would you expect to get correct? (Hint: This expectation is just the sum of probabilities of guessing correcting on the first, second, third, and fourth draws.) How does this compare with the number you would expect if draws were made with replacement? What are the implications of these results on the experiment conducted with Olof Jonsson? e) According to the researchers, the computer randomly selected the precognition mode with equal probability for each run of 24 trials. There 12 runs in the experiment. From the table, we can see that 3 of the runs (72 trials / 24 trials per run) were in precognition mode. How likely is it to get as few as 3 runs out of 12 in precognition mode? Comment on the experimental results in view of your findings.