EXAMPLE 94

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Increasing the Sample Size

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Suppose now that the experimenter in the previous example has 100 bulbs with which to experiment We could work out all the probabilities for the 101 possible values for X and

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9

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Statistical Inference I

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would most certainly use a computer to do this For this sample size, the binomial distribution is very normal-like with a maximum at the expected value, np, and standard deviation n p (1 p) In either case here, n = 100 If p = 050, we the probability distribution nd centered about np = 100 050 = 50 with standard deviation 100 050 050 = 5 Since we are seeking a critical region in the upper tail of this distribution, we look at values of X at least one standard deviation from the mean, so we start at X = 55 We show some probabilities in Table 92 Table 92 Critical region X 56 X 57 X 58 X 59 X 60 X 61 X 62 01356 00967 00443 00284 00176 00105 00060 00000 00000 00001 00001 00003 00007 00014

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We see that decreases as we move to the right on the probability distribution and that increases We have suggested various critical regions here and determined the resulting values of the errors This raises the possibility that the size of one of the errors, say , is chosen in advance and then a critical region found that produces this value of The consequence of this is shown in Table 92 It is not possible, for example, to choose = 005 and nd an appropriate critical region This is because the random variable is discrete in this case If the random variable were continuous, then it is possible to specify in advance We show how this is so in the next example

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EXAMPLE 95

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Breaking Strength

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The breaking strength of steel wires used in elevator cables is a crucial characteristic of these cables The cables can be assumed to come from a population with known = 400 lb Before accepting a shipment of these steel wires, an engineer wants to be con dent that >10,000 lb A sample of 16 wires is selected and their mean breaking strength X is measured It would appear sensible to test the null hypothesis H0 : = 10,000 lb against the alternative Ha : < 10,000 A test will be based on the sample mean, X The central limit theorem tells us that X N In this case, we have X N 400 , 16 = N ( , 100) , n

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If the critical region has size 005, so that = 005, then we would select a critical region in the left tail of the normal curve The situation is shown in Figure 92

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and the Power of a Test

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Distribution plot

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Normal, Mean = 10,000, StDev = 100 0004

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0003 Density

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005 0000 9836 10,000 X

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The value of shaded area in Figure 92 is 005, so the z score is 1645 This means that the critical value of X is 1645 = (x 10,000)/(100) or x = 10000 1645 = 98355 So the null hypothesis should be rejected if the sample mean is less than 98355 lb

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AND THE POWER OF A TEST

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What is , the size of the Type II error in this example We recall that = P(H0 is accepted if it is false) or = P(H0 is accepted if the alternative is true) We could calculate easily in our rst example since in that case we had a speci c alternative to deal with (namely, p = 075) However, in this case, we have an in nity of alternatives ( < 10,000) to deal with The size of depends upon which of these speci c alternatives is chosen We will show some examples We use the notation alt to denote the value of when a particular alternative is selected First, consider 9800 = P(X > 98355 if = 9800) =P Z> 98355 9800 = 0355 100 = 0361295

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