#  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.

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 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.

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 1. 2. 3.

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