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CHAPTER 21 More About Tests: “What Can Go Wrong?”

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P-values - The probability that the test statistic would have a value as extreme or more than what is actually observed if the null hypothesis is true In other words... is it far out in the tails of the distribution?

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Level of significance - Is the amount of evidence necessary before we begin to doubt that the null hypothesis is true Is the probability that we will reject the null hypothesis, assuming that it is true Denoted by –Can be any value –Usual values: 0.1, 0.05, 0.01 –Most common is 0.05 (default value)

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When a hypothesis test is conducted, α must be chosen ahead of time, lest you be accused of fudging your results to say what you want them to say.

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Statistically significant – as smallsmallerThe p-value is as small or smaller than the level of significance ( ) fail to rejectIf p > , “fail to reject” the null hypothesis at the level. rejectIf p < , “reject” the null hypothesis at the level.

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Facts about p-values: ALWAYS make decision about the null hypothesis! Large p-values show support for the null hypothesis, but never that it is true! Small p-values show support that the null is not true. Double the p-value for two-tail (=) tests Never acceptNever accept the null hypothesis!

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When you perform a hypothesis test you make a decision: When you make one of these decisions, there is a possibility that you could be wrong! That you made an error! reject H 0 or fail to reject H 0

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There are two decisions that we make; reject or fail to reject. Each could possibly be a wrong decision; therefore, there are two types of errors.

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Type I error When you reject the null hypothesis that is really true i.e. Reject when you should have failed to reject The probability of a type I error is

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Type II error When you fail to reject the null hypothesis when it is false i.e. fail to reject when you should have rejected The probability of a type II error is

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The probability of correctly rejecting a false null hypothesis. Power is the probability that the test will do what it is supposed to do i.e. reject the null hypothesis when in fact it should be rejected. Power Note: We will not be calculating power, but you will need to understand what affects the power of a test.

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H 0 TrueH 0 False Reject Fail to reject Type I error Correct Type II error Suppose H 0 is true & we fail to reject it, what type of decision was made? Suppose H 0 is false & we reject it, what type of decision was made? Suppose H 0 is true & we reject it, what type of decision was made? Suppose H 0 is false & we fail to reject it, what type of decision was made? POWER

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Facts: Every time you make a decision, you have potentially made an error. & are inversely related 00 aa Fail to reject H 0 Reject H 0 As decreases, increases As increases, decreases

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Facts continued: The seriousness of the error types is determined by the specific situations. –Depending upon the situation type I or type II may be the more serious. DO NOTWe often DO NOT know if an error is made in real life. –Except for cases like Firestone tires http://www.firestone-tire- recall.com/pages/overview.htmlhttp://www.firestone-tire- recall.com/pages/overview.html Drugs like Fen-phen http://www.lawyersandsettlements.com/lawsuit/fenphen.html Someone made an error with these products, with deadly consequences

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Let’s look at a jury trial. a)State the hypotheses H 0 = the defendant is found not guilty H a = the defendant is found guilty b) What type of error is committed when a guilty person is found not guilty? Type II…you have set the guilty free c) What type of error is committed when an innocent person is found guilty? Type I d) Which of the two type of errors do you believe our justice system considers to be the more serious mistake? Type I – an innocent person is judged guilty This is why we have the appeals process. e) State the power in context of the problem? The probability that we find a guilty person guilty.

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Lay’s Chip Company decides to accept a truck load of potatoes based upon the result obtained form a sample of potatoes from the truck load. a)State the null and alternative hypotheses in words. H 0 = potatoes are good (we keep them) H a = potatoes are not good (we send them back) b) Identify a Type I error in context of the problem. We determined the potatoes were bad when they were actually good and we sent them back The supplier would likely lose money by having to buy more potatoes when in fact it wasn’t necessary to do so. c) Explain a consequence of the Type I error.

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Lay’s Chip Company decides to accept a truck load of potatoes based upon the result obtained form a sample of potatoes from the truck load. d) Identify a Type II error in context of the problem. We failed to identify bad potatoes and kept them a)State the null and alternative hypotheses in words. H 0 = potatoes are good (we keep them) H a = potatoes are not good (we send them back) By keeping bad potatoes we could create a bad product e) Explain a consequence of the Type II error.

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Lay’s Chip Company decides to accept a truck load of potatoes based upon the result obtained form a sample of potatoes from the truck load. a)State the null and alternative hypotheses in words. H 0 = potatoes are good (we keep them) H a = potatoes are not good (we send them back) Type II – keeping the bad potatoes f) Which consequence do you consider the more serious error (if you were the company)? g) State the power in context of the problem. The probability that we correctly identified bad potatoes as bad and sent them back.

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A doctor is considering a new medication to help fight infections. However, the medication has the possibility of being highly toxic to the patient. You will test the medication to determine toxicity. What are the hypotheses? What are the Type I & II errors? Which is more serious? H 0 : medicine is not toxic H a : medicine is toxic Type I: say medicine is toxic when it really isn’t Type II : say medicine isn’t toxic when it really is Most would consider a type II error more serious since people could be harmed.

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