Type I and II Errors Power

 Better Batteries a) What conclusion can you make for the significance level α = 0.05? b) What conclusion can you make for the significance level α = 0.01? Since the P -value, , is less than α = 0.05, we reject H 0 and conclude that the company’s deluxe AAA batteries last longer than 30 hours, on average. When we reject H 0, we call the result statistically significant. A company has developed a new deluxe AAA battery that is supposed to last longer than its regular AAA battery. However, these new batteries are more expensive to produce, so the company would like to be convinced that they really do last longer. Based on years of experience, the company knows that its regular AAA batteries last for 30 hours of continuous use, on average. The company selects an SRS of 15 new batteries and uses them continuously until they are completely drained. A significance test is performed using the hypotheses H 0 : µ = 30 hours H a : µ > 30 hours where µ is the true mean lifetime of the new deluxe AAA batteries. The resulting P- value is Since the P -value, , is greater than α = 0.01, we fail to reject H 0. Therefore, we cannot conclude that the deluxe AAA batteries last longer than 30 hours, on average.

 Just like our court system, we will make the wrong decision from time to time.  What are the two ways the court can make the wrong decision?  Decide the defendant is not guilty when in fact the defendant is guilty.  Decide the defendant is guilty when in fact the defendant is not guilty.  What is the wording we use in our significance tests? We either…  Reject H 0.  Fail to reject H 0.

Truth about the population H 0 true H 0 false (H a true) Conclusion based on sample Reject H 0 Type I error Correct conclusion Fail to reject H 0 Correct conclusion Type II error Type I Error: Reject the H 0 when the H 0 is true. Type II Error: Fail to reject the H 0 when the H 0 is false.

Truth about the population H 0 true H 0 false (H a true) Conclusion based on sample Reject H 0 Type I error Correct conclusion Fail to reject H 0 Correct conclusion Type II error This smiley face has a name. It’s called POWER. We control the probability of making a Type I error by using a low α. The probability of making a Type II error is called β.

 A company has developed a new deluxe AAA battery that is supposed to last longer than its regular AAA battery. However, these new batteries are more expensive to produce, so the company would like to be convinced that they really do last longer. Based on years of experience, the company knows that its regular AAA batteries last for 30 hours of continuous use, on average. The company selects an SRS of 15 new batteries and uses them continuously until they are completely drained. A significance test is performed using the hypotheses  H 0 : µ = 30 hours  H a : µ > 30 hours  where µ is the true mean lifetime of the new deluxe AAA batteries. The resulting P-value is  A) Describe a Type I Error in the context of this problem.  B) Describe a Type II Error in the context of this problem.  C) If you had to choose one of the standard levels of α for this test, would you choose α = 0.10, 0.05, or 0.01? Justify your answer.  Which do you think is a more serious error in this case, Type I or Type II?

 Memorize Type I and Type II Errors in words.  Memorize Power in words.  The probability of making a Type I error is called α. Lower levels of α mean we are less likely to make a Type I error.  The probability of making a Type II error is called β. We can’t lower both α and β at the same time, so we generally choose to control α.  There are a couple of ways to increase power.  Increase sample size.  Use a higher level for α.

 P(Type II error) = 1 – Power.  If you haven’t already discovered the summary pages in your book, they are wonderful. Section 9.1 is summarized on p. 545.