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### Are you Psychic (or just psychotic)?

Do you have hidden psychic powers? Is there a career waiting for you on the Psychic Hotline? We will use statistical hypothesis testing to find out. (Disclaimer--we actually are more interested in learning how statistical hypothesis testing works, but I hope you find the exercise both enjoyable and instructive.)

You will pair up, one of each pair taking the role of the sender and the other of the receiver. You will then sit back-to-back, and the sender will write a circle or an X on a sheet of paper. He/She will then concentrate on that image, and the receiver will write what he/she thinks it is. The receiver then passes the result to the sender who tallies a score showing number right and number wrong. You will do this procedure 25 times.

However, before we get to this step, we have to decide what we are looking for. If there is nothing psychic happening, we expect that by random chance about 50% of the answers will be correct.

We need to begin with an assumption, and it will be the assumption that we want to disprove. Should we begin by assuming you are not psychic? Or should we begin with the assumption that you are in fact psychic? Explain.

(Should we consider all wrong answers as evidence of some sort of psychic ability? Or do we only want to conclude a person is psychic when most answers are correct? This is asking whether we want a one-tailed test or a two-tailed test.)

We will then gather data and ask whether these results are consistent with the hypothesis. For example, if you get 14 right of 25 tries, you might argue that since you did better than expected, this result indicates you have a little bit of psychic ability. However, most people would say that you just got lucky, since guessing 13 or 14 will happen a lot by random chance if you are not psychic. On the other hand, if you guess 25 of 25 correct, that also could happen by random chance even if you are not psychic, but it is very, very unlikely. Instead of explaining this result as luck, we might conclude that these results are highly suggestive that you are psychic.

Before you begin to gather data, you must specify what evidence will make you reject the hypothesis that you are not psychic and accept instead the alternative that you are.

How unlikely will the results have to be before you will think psychosis (oops, I guess that should be psychic ability) is plausible? 1 in 20? 1 in 100? 1 in 1000? 1 in 1000000?

If you are not psychic, you should get 50% right by random chance. We are drawing from a zero-one box. For the box the mean is .5 and the standard deviation is .5. If we compute what we expect for the sum, the expected value of 25 draws is 12.5 and the standard error of the sum is 2.5. (If you do not know where these numbers came from, please stop and ask.) With this we can do a z-test:

((Observed number right)-12.5)/2.5 =

What do you decide? Is it chance, or are you psychic? Explain.

### Are you Psychic (or just psychotic)?

Do you have hidden psychic powers? Is there a career waiting for you on the Psychic Hotline? We will use statistical hypothesis testing to find out. (Disclaimer--we actually are more interested in learning how statistical hypothesis testing works, but I hope you find the exercise both enjoyable and instructive.)

You will pair up, one of each pair taking the role of the sender and the other of the receiver. You will then sit back-to-back, and the sender will write a circle or an X on a sheet of paper. He/She will then concentrate on that image, and the receiver will write what he/she thinks it is. The receiver then passes the result to the sender who tallies a score showing number right and number wrong. You will do this procedure 20 times.

However, before we get to this step, we have to decide what we are looking for. If there is nothing psychic happening, we expect that by random chance about 50% of the answers will be correct. In fact, using the binomial tables, we can compute exact probabilities of what random guessing will yield:

Probability of 10 right = .1762
Probability of 9 thru 11 right = .4965
Probability of 8 thru 12 right = .7368
Probability of 7 thru 13 right = .8847
Probability of 6 thru 14 right = .9586
Probability of 5 thru 15 right = .9882
Probability of 4 thru 16 right = .9974
Probability of 3 thru 17 right = .9995
Probability of 2 thru 18 right = .9999

We can present this information in this way:
Probability of being 1 or more away by random chance = .8238
Probability of being 2 or more away by random chance = .5035
Probability of being 3 or more away by random chance = .2732
Probability of being 4 or more away by random chance = .1253
Probability of being 5 or more away by random chance = .0414
Probability of being 6 or more away by random chance = .0118
Probability of being 7 or more away by random chance = .0026
Probability of being 8 or more away by random chance = .0005
Probability of being 9 or more away by random chance < .0001

Probability of getting 0-9 or 11-20 by random chance = .8238
Probability of getting 0-8 or 12-20 by random chance = .5035
Probability of getting 0-7 or 13-20 by random chance = .2732
Probability of getting 0-6 or 14-20 by random chance = .1253
Probability of getting 0-5 or 15-20 by random chance = .0414
Probability of getting 0-4 or 16-20 by random chance = .0118
Probability of getting 0-3 or 17-20 by random chance = .0026
Probability of getting 0-2 or 18-20 by random chance = .0005
Probability of getting 0-1 or 19-20 by random chance < .0001

We need to begin with an assumption, and it will be the assumption that we want to disprove. Should we begin by assuming you are not psychic? Or should we begin with the assumption that you are in fact psychic? Explain.

(Should we consider all wrong answers as evidence of some sort of psychic ability? Or do we only want to conclude a person is psychic when most answers are correct? How would you chance the tables if you only want to consider correct answers?)

We will then gather data and ask whether these results are consistent with the hypothesis. For example, if you get 11 right of 20 tries, you might argue that since you did better than expected, this result indicates you have a little bit of psychic ability. However, most people would say that you just got lucky, since guessing 11 will happen a lot by random chance if you are not psychic. On the other hand, if you guess 20 of 20 correct, that also could happen by random chance even if you are not psychic, but it is very, very unlikely. Instead of explaining this result as luck, we might conclude that these results are highly suggestive that you are psychic.

Before you begin to gather data, you must specify what evidence will make you reject the hypothesis that you are not psychic and accept instead the alternative that you are.