Question

Every time you conduct a hypothesis test, there are four possible outcomes of your decision to reject or not reject the null hypothesis: (1) You don't reject the null hypothesis when it is true, (2) you reject the null hypothesis when it is true, (3) you don't reject the null hypothesis when it is false, and (4) you reject the null hypothesis when it is false. Consider the following analogy: You are a juror in a murder trial. For every defendant, you must decide whether to convict the defendant based on your assessment of whether she committed the crime as charged. Suppose your null hypothesis is that the defendant is innocent. As in hypothesis testing, there are four possible outcomes of your decision: (1) You do not convict the defendant when the defendant is innocent, (2) you convict the defendant when the defendant is innocent, (3) you do not convict the defendant when the defendant is guilty, and (4) you convict the defendant when the defendant is guilty. Which of the following outcomes corresponds to a Type I error? You convict the defendant when the defendant is guilty. You convict the defendant when the defendant is innocent. You do not convict the defendant when the defendant is innocent. You do not convict the defendant when the defendant is guilty. Which of the following outcomes corresponds to a Type II error? You do not convict the defendant when the defendant is innocent. You do not convict the defendant when the defendant is guilty. You convict the defendant when the defendant is guilty. You convict the defendant when the defendant is innocent. As a juror, the worst error you can make is to convict the defendant when the defendant is innocent. The probability that you make this error, in this hypothesis testing analogy, is described by . To complete the hypothesis test, you must select an alpha level. A smaller alpha the size of the critical region and the chance of making a Type I error.

Answer 1

Null Hypo: Defendant is innocent

Alternative hypo: Defendant is not innocent.

Therefore if we are rejecting the null hypothesis the defendant should be convicted.

But if null hypothesis is true the defendant should not be convicted.

Type 1 error: Rejecting null hypothesis when it is true.

According to our case

(a) Type 1 error:  you convict the defendant when the defendant is innocent(not guilty)

Explanation: Reject null hypo - Defendant proved guilty and is convicted

Null Hypo is true - Defendant is innocent.

(b) Type 2 error: Not rejecting null hypothesis when it is false.

According to our case

Type 2 error:  You do not convict the defendant when the defendant is guilty

Explanation: Not rejecting null Hypo - Defendant proved innocent and is not convicted

Null Hypo is false - Defendant is guilty.

(c)  As a juror the worst decision is to convict a defendant when not guilty. As seen above it is the type 1 error that is worst error a juror can make.

Probability of Type 1 error is described by

(d) The question is not clear but I will provide answers with best of my knowledge.

The level is pre decided mostly since this provides the decision criteria whether the p- value is greater than or smaller than . A smaller reduces the chance of making type 1 error. Also larger sample can also help to reduce the type 1 error. because the results will be more accurate.