Do you have ESP? A researcher looking for evidence of extrasensory perception (ESP) tests $500$ subjects. Four of these subjects do significantly better $(P<0.01)$ than random guessing.

a. Is it proper to conclude that these four people have ESP? Explain your answer.

b. What should the researcher now do to test whether any of these four subjects have ESP?

$$\begin{gathered} n=500 \\ x=4 \\ P<0.01 \end{gathered}$$

No, since a $p-$value of $0.01$ would be expected in around $1\%$ of the $500$subjects Because $1\%$ of $500$ subjects corresponds to $5$ subjects, it's likely that four of the $500$ subjects have a $p-$value of less than $0.01$

When null hypotheses of random guessing are true, the $p-$value is really the probability of getting test findings, or more extreme, when null hypotheses of random guessing are true. As a result, it's likely that these four participants with p-values less than $0.01$did not have EPSP.

Thus, it is possible that 4 of the $500$ subjects have a $p-$value below the $0.01$

These four subjects should be retested for ESP. If they perform much better than random guessing once more, they are highly likely to have ESP.

Thus, Researchers should retest these $4$ subjects for ESP.