An ESP experiment is done in which a participant guesses which of 8 cards the researcher has randomly picked, where each card is equally likely to be selected. This is repeated for 200 trials. The null hypothesis is that the subject is guessing, while the alternative is that the subject has ESP and can guess at higher than the chance rate. Write out the type 1 and type 2 errors in terms of this problem.

Type I errors result when you reject the null and it's true. Type II errors result when you accept the null and it's false.

If you reject the null hypothesis that the subject is guessing and it's true, you have made a Type I error. If you accept the null that the subject is guessing and it's false, you have made a Type II error.

In hypothesis testing, we evaluate the null hypothesis and compare it with the alternative hypothesis. In this case, the null hypothesis is that the participant is guessing, while the alternative hypothesis is that the participant has ESP and can guess at a higher rate than chance.

Type 1 Error:
A Type 1 error occurs when we reject the null hypothesis when it is actually true. In the context of this problem, a Type 1 error would happen if we conclude that the participant has ESP and can guess at a higher rate than chance when, in reality, they are just guessing.

Type 2 Error:
A Type 2 error occurs when we fail to reject the null hypothesis when it is actually false. In the context of this problem, a Type 2 error would occur if we fail to conclude that the participant has ESP and can guess at a higher rate than chance when, in reality, they do possess ESP.

To determine the type 1 and type 2 errors more explicitly, we need to establish a significance level (often denoted as α) for the hypothesis test. The significance level indicates how unlikely we are willing to accept a Type 1 error. Common values for α are 0.05 or 0.01, indicating a 5% or 1% chance, respectively.

Let's consider the outcomes in terms of the null and alternative hypotheses:

Type 1 Error (False Positive):
- Reject the null hypothesis (Participant is guessing).
- The participant is actually just guessing.

Type 2 Error (False Negative):
- Fail to reject the null hypothesis (Participant is guessing).
- The participant actually has ESP and can guess at a higher rate than chance.

It is important to note that we cannot entirely eliminate both types of errors. The significance level and the data at hand influence the likelihood of committing each error.