 When we perform a hypothesis test, we make a decision to either Reject the Null Hypothesis or Fail to Reject the Null Hypothesis.  There is always the possibility that we made an incorrect decision.  We can make an incorrect decision in two ways:

1. If we reject the null when in fact the null is true, this is a Type I error 2. If we fail to reject the null when in fact the null is false, this is a Type II error.

 A Type I error is the mistake of rejecting the null hypothesis when it is true.  In testing for a medical disease, the null hypothesis is usually the assumption that a person is healthy. A Type I error is a false positive; a healthy person is diagnosed with the disease.  The probability of rejecting the null hypothesis when it is true is equal to the alpha-level of the test.  If alpha =.05, in the long-run, we will incorrectly reject the null hypothesis when it is really true about 5% of the time.

 A type II error is made when we fail to reject the null hypothesis when it is false and the alternative is true. Prob(Type II Error) =  In medical testing for a disease, this would be equivalent to a person who has the disease being diagnosed as disease free. This is called a false negative.  Our ability to detect a false hypothesis is called the power of the test. The Power of a Test is the probability that it correctly rejects a false null hypothesis. Power = 1 - Prob (Type II Error) or

Ex: Let, and. Describe both types of error and their consequences.

Time for a worksheet!

A new car had been released for which the manufacturer reports that the car gets 23mpg for city driving. A consumer group feels that the true mileage is lower and performs the following hypothesis test, using α =.05. Assumptions: The gas mileage in the sample of 30 cars comes from an SRS drawn from the population of all mileage and the standard deviation is known, δ = 1.2. Null and alternate hypothesis: What is a Type I error in the context of this problem? What is a Type II error in the context of this problem?

The Probability of Type I error is, ! The Probability of Type II error is, and is dependent upon the following

Ex: the mean salt content of a certain brand of potato chip is supposed to be 2 mg. The salt contents vary normally with standard deviation.1 mg. For each batch produced, an inspector takes a sample of 5 chips and measures the salt content. The entire batch is rejected if the mean salt content of the sample is significantly higher than 2mg, at the 5% significance level. The company has decided that having a mean salt content that is.05mg higher than believed is entirely unacceptable. Name the null and alternate hypothesis. Explain the meaning of a type I error in this problem. Find the probability of a type I error. Rule:

Suppose that a baseball player who has always been a.250 career hitter suddenly improves over one winter to the point where the probability of getting a hit during an at-bat is.333. He asks management to renegotiate his contract, since he is a more valuable player now. Management has no reason to believe that he is better than a.250 hitter. Suppose they decide to give the player 20 at-bats to show that his true batting average is greater than.250.

 H(o): p =.250 H(a): p >.250  If alpha is set at 0.05, and the player is given 20 at-bats to show that he has improved, what does his average need to be after those 20 at-bats in order to reject the null hypothesis?

If he is really a.333 hitter, how often will he have the necessary average of about.410 with n=20 at-bats, necessary to reject the null hypothesis?

Alpha = 0.05 Null Hypothesis p = Alternative Hypothesis The Truth p= Retain Null Type II Error Reject Null Power

The value is the power of the test to detect the change. In other words, approximately 22.5% of the time, we will be able to correctly reject the null hypothesis that the batter is a.250 hitter, in favor of the alternative hypothesis (if the player has a.333 batting average).

 A type II error is the probability of making the incorrect decision to fail to reject the null hypothesis when it is false. In this case the probability of a type II error is To calculate the probability of a Type II error, you need to state an alternative value, in this case p=0.333  Power = 1 - Prob(Type II Error) Power = 1 - beta

The is the probability of Ways to increase