1. a. b. c. d. Yes. It is an assertion about the value of a parameter. No. The sample median
~ X is not a parameter.
No. The sample standard deviation s is not a parameter. Yes. The assertion is that the standard deviation of population #2 exceeds that of population #1 No.
X and Y are statistics rather than parameters, so cannot appear in a hypothesis.Yes. H is an assertion about the value of a parameter.
2. a. b. c. d. These hypotheses comply with our rules. Ho is not an equality claim (e.g.
σ = 20 ), so these hypotheses are not in compliance.
Ho should contain the equality claim, whereas Ha does here, so these are not legitimate. The asserted value of µ1 conditions are not met.
− µ 2 in Ho should also appear in Ha. It does nothere, so our
Each S2 is a statistic, so does not belong in a hypothesis. We are not allowing both Ho and Ha to be equality claims (though this is allowed in more comprehensive treatments of hypothesis testing). These hypotheses comply with our rules. These hypotheses are in compliance.
In this formulation, Ho states the welds do not conform to specification. Thisassertion 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: µ < 100 results 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
4. When thealternative is Ha: µ < 5 , 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 (Ho 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 aserious 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: µ > 5 , the type II error (now stating that the water is safe when it isn’t) is the more serious of the two errors. Let σ denote the population standard deviation. The appropriate hypothesesare
H o : σ = .05 vs H a : σ < .05 . 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 Ho can 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.
H o : µ = 40 vs H a : µ ≠ 40 , where µ is the true average burn-out amperage for this type of fuse. The alternative reflects the fact that a departure from µ = 40 in either
direction is of concern. Notice that in this formulation, it is initially believed that the value of µ is the design value of 40.
A type I error here involves saying that the plant is not in compliancewhen in fact it is. A type II error occurs when we conclude that the plant is in compliance when in fact it isn’t. Reasonable people may disagree as to which of the two errors is more serious. If in your judgement it is the type II error, then the reformulation H o : µ = 150 vs H a : µ < 150 makes the type I error more serious.
µ1 = the average amount of warpage for the regular laminate,and µ 2 = the analogous value for the special laminate. Then the hypotheses are H o : µ1 = µ 2 vs H o : µ1 > µ 2 .
Let Type I error: Conclude that the special laminate produces less warpage than the regular, when it really does not. Type II error: Conclude that there is no difference in the two laminates when in reality, the special one produces less warpage.
Chapter 8: Tests of...