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. Xand Yare 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. Hois not an equality claim (e.g. 20=s), so these hypotheses are not in compliance. c. Hoshould contain the equality claim, whereas Hadoes here, so these are not legitimate. d. The asserted value of 21mm-in Hoshould also appear in Ha. It does not here, so our conditions are not met. e. Each S2is a statistic, so does not belong in a hypothesis. f. We are not allowing both Hoand Hato 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, Hostates 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 Ha: 100<mresults in the welds being believed in conformance unless provided otherwise, so the burden of proof is on the non-conformance claim.

Chapter 8: Tests of Hypotheses Based on a Single Sample 238 4. When the alternative is Ha: 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 Ho) when it isn’t (Hois 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 Ha: 5m, 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 sdenote the population standard deviation. The appropriate hypotheses are 05.:=soHvs 05.:<saH. 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 Hocan be rejected in favor of Ha. 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. 40:=moHvs 40:≠maH, where mis the true average burn-out amperage for this type of fuse. The alternative reflects the fact that a departure from 40=min either direction is of concern. Notice that in this formulation, it is initially believed that the value of mis the design value of 40.