Chapter 3 Investigating Independence Objectives Students will be able to: 1) Understand what it means for attempts to be independent 2) Determine when evidence in a hypothesis test is statistically significant 3) Compare and contrast Type I and Type II errors

Terminology Time!!! An athlete’s attempts are independent if his or her ABILITY to be successful is the same after a successful PERFORMANCE and after an unsuccessful PERFORMANCE. For example, Kyle is a basketball player. His shot attempts are independent if the success of one shot does not depend on the success of a previous shot. Another example: Michelle’s attempts at flipping a coin and wanting it to land on heads are independent if the success of one flip landing on heads does not depend on the success of a previous flip.

Refresher… The significance level of a test is a predetermined level of evidence that is required to essentially rule out RANDOM CHANCE as a plausible explanation. We say evidence in a hypothesis test is statistically significant whenever the evidence is convincing enough to reject the null hypothesis. “Statistically significant” is essentially a synonym for “convincing.”

Example: Suppose that we tested whether a basketball player had a greater ABILITY to make a free-throw following a made free-throw than following a missed free-throw and that the results of the test were not statistically significant. What does it mean that the results of the test were not statistically significant? This means we do not have convincing evidence to support the claim that the player has a greater ABILITY to make a free-throw following a made free-throw than following a missed one. Note: If the results were statistically significant, then we would have convincing evidence to support the alternative hypothesis.

Type I and Type II Errors Even if we perform all of the correct procedures in a hypothesis test, we can still make an error with our conclusion. The different kinds of errors we can make are known as Type I and Type II errors.

Type I Error A Type I error occurs when we find convincing evidence that an alternative hypothesis is true, when it reality it is not true. (This would lead us to reject the null hypothesis). A Type I error is also known as a “false positive.” Examples of Type I errors: In a courtroom, the null hypothesis is that the defendant is not guilty and the alternative hypothesis is that the defendant is guilty. A Type I error would be convicting the defendant, when in reality he is not guilty. You go to the doctor for a checkup. The doctor tells you that you are sick, when in reality you are not.

Type II Error A Type II error occurs when we do not find convincing evidence that an alternative hypothesis is true, when in reality it is true. (This would lead us to fail to reject the null). A Type II error is also known as a “false negative.” Example of Type II errors: In a courtroom, a Type II error would be not convicting a defendant, when in reality he is guilty. You go to the doctor for a checkup. The doctor tells you that you are not sick, when in reality you are sick.

Mr. Chart best summarizes the types of errors that can be made:

Type I errors occur because we go into the hypothesis testing process willing to accept a certain amount of risk. If we have a 5% level of significance, then we can expect to make a Type I error about 5% of the time. Type I errors can be reduced by using a smaller significance level. This would require evidence to be more convincing before concluding that the alternative hypothesis is true. Caution: You do not want to make the significance level too small. While it will decrease the likelihood of causing a Type I error, it will also decrease the likelihood that we decide to support the alternative hypothesis (even when it is actually true!). That would lead to an increase in causing Type II errors.

Type II errors occur when the number of PERFORMANCES is small. Remember that very unusual PERFORMANCES can happen just by RANDOM CHANCE. Type II errors can be reduced by increasing your sample size. By gathering more data, we can be more confident that an athlete’s PERFORMANCE will be closer to his or her actual ABILITY.

Super fun example time!!! Suppose that we performed a hypothesis test to see if the Los Angeles Dodgers had a greater ABILITY to win following a win than following a loss. a) State the hypotheses we are interested in testing.

b) Describe a Type I error and Type II error in the context of this question. Reminder: A Type I error is finding convincing evidence that an alternative hypothesis is true, when in reality it is not. A Type II error is not finding convincing evidence that an alternative hypothesis is true, when in reality it is. A Type I error would be if we were convinced the Dodgers had a greater ABILITY to win following a win than following a loss when, in fact, the team’s ABILITY was the same. A Type II error would be if we aren’t convinced that the Dodgers’ ABILITY to win was greater following a win when, in fact, it was.

c) If the results of the test were statistically significant, which type of error could we have committed? If the results of the test were statistically significant, then we would have rejected the null hypothesis. Therefore, we could have committed a Type I error (finding convincing evidence that an alternative hypothesis is true, when in reality it is not; thus leading us to reject the null).