Chapter 8 - CHAPTER 8 Section 8.1 1. a. b. c. d. Yes. It is...

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237 CHAPTER 8 Section 8.1 1. a. Yes. It is an assertion about the value of a parameter. b. No. The sample median X ~ is not a parameter. c. No. The sample standard deviation s is not a parameter. d. Yes. The assertion is that the standard deviation of population #2 exceeds that of population #1 e. No. X and Y are statistics rather than parameters, so cannot appear in a hypothesis. f. Yes. H is an assertion about the value of a parameter. 2. a. These hypotheses comply with our rules. b. H o is not an equality claim (e.g. 20 = s ), so these hypotheses are not in compliance. c. H o should contain the equality claim, whereas H a does here, so these are not legitimate. d. The asserted value of 2 1 m m - in H o should also appear in H a . It does not here, so our conditions are not met. e. Each S 2 is a statistic, so does not belong in a hypothesis. f. We are not allowing both H o and H a to be equality claims (though this is allowed in more comprehensive treatments of hypothesis testing). g. These hypotheses comply with our rules. h. These hypotheses are in compliance. 3. In this formulation, H o states the welds do not conform to specification. This assertion will not be rejected unless there is strong evidence to the contrary. Thus the burden of proof is on those who wish to assert that the specification is satisfied. Using H a : 100 < m results in the welds being believed in conformance unless provided otherwise, so the burden of proof is on the non-conformance claim.
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Chapter 8: Tests of Hypotheses Based on a Single Sample 238 4. When the alternative is H a : 5 < m , the formulation is such that the water is believed unsafe until proved otherwise. A type I error involved deciding that the water is safe (rejecting H o ) when it isn’t (H o is true). This is a very serious error, so a test which ensures that this error is highly unlikely is desirable. A type II error involves judging the water unsafe when it is actually safe. Though a serious error, this is less so than the type I error. It is generally desirable to formulate so that the type 1 error is more serious, so that the probability of this error can be explicitly controlled. Using H a : 5 m , the type II error (now stating that the water is safe when it isn’t) is the more serious of the two errors. 5. Let s denote the population standard deviation. The appropriate hypotheses are 05 . : = s o H vs 05 . : < s a H . With this formulation, the burden of proof is on the data to show that the requirement has been met (the sheaths will not be used unless H o can be rejected in favor of H a . Type I error: Conclude that the standard deviation is < .05 mm when it is really equal to .05 mm. Type II error: Conclude that the standard deviation is .05 mm when it is really < .05. 6.
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This note was uploaded on 07/20/2009 for the course MATH 3502 taught by Professor Zahra during the Summer '09 term at Carleton CA.

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Chapter 8 - CHAPTER 8 Section 8.1 1. a. b. c. d. Yes. It is...

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