Question

This problem is designed to give you an understanding of the methodology behind hypothesis testing. Ever wonder how someone in America can be arrested if they really are presumed innocent, why a defendant is found not guilty instead of innocent, or why Americans put up with a justice system which sometimes allows criminals to go free on technicalities? These questions can be understood by understanding the similarity of the American justice system to hypothesis testing in statistics and the two types of errors it can produce. True, the trial process does not use numerical values as does hypothesis testing in statistics, but both the justice system and statistical versions of hypothesis testing share common elements.

a. In the American justice system, a defendant is always "innocent until proven guilty." Suppose you are the prosecutor in a trial. What would your null and alternative hypotheses be in this case? State them in words (there are no numerical values here!).

b. The two possible verdicts that a jury can return are "guilty" and "not guilty." How do these decisions correspond to the two possible decisions in hypothesis testing?

c. If an innocent person has been found guilty, what type of hypothesis testing error has been made?

d. If a guilty person is found not guilty in his/her trial, what type of hypothesis testing error has been made?

e. In the justice system and statistics there is no possibility of absolute proof, and so a standard has to be set for rejecting the null hypothesis. In the justice system the standard is "a reasonable doubt." The null hypothesis has to be rejected beyond a reasonable doubt. In statistics the standard is the maximum acceptable probability of incorrectly rejecting a true null hypothesis. What is the name of this element in a statistical hypothesis test?

Answer 1

Explanation:

Hello!

a. In the American justice system, a defendant is always "innocent until proven guilty." Suppose you are the prosecutor in a trial. What would your null and alternative hypotheses be in this case? State them in words (there are no numerical values here!).

H₀: the defendant is innocent.

H₁: the defendant is guilty.

In the null hypothesis, it is always stated the "no change" premise*, that is why it always carries the = sign.

*For example Presumably known information about the population of study that is accepted until finding statistical evidence that proves it to be false.

b. The two possible verdicts that a jury can return are "guilty" and "not guilty." How do these decisions correspond to the two possible decisions in hypothesis testing?

There are two possible decisions that can be made when conducting a hypothesis analysis:

To reject the null hypothesis: this is equal to the jury finding the defendant guilty.

To not reject the null hypothesis: This is equal to the jury finding the defendant not guilty.

c. If an innocent person has been found guilty, what type of hypothesis testing error has been made?

If the person is "innocent" that means the null hypothesis is true.

Finding the person "guilty" means that the null hypothesis was rejected.

Rejecting a true null hypothesis is a Type I error.

d. If a guilty person is found not guilty in his/her trial, what type of hypothesis testing error has been made?

If the person is "guilty" that means the null hypothesis is false.

Finding the person "not guilty" is equal to not rejecting the null hypothesis.

Not rejecting a false null hypothesis is a Type II error.

e. In the justice system and statistics, there is no possibility of absolute proof, and so a standard has to be set for rejecting the null hypothesis. In the justice system, the standard is "a reasonable doubt." The null hypothesis has to be rejected beyond a reasonable doubt. In statistics, the standard is the maximum acceptable probability of incorrectly rejecting a true null hypothesis. What is the name of this element in a statistical hypothesis test?

The name of this statistical element is "Significance level" symbolized α and it is the probability of committing a Type I error.

I hope it helps!