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# Week12 - STA 2023 c B.Presnell& D.Wackerly Lecture 19...

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Unformatted text preview: STA 2023 c B.Presnell & D.Wackerly - Lecture 19 Thought: Eagles may soar, but weasels aren’t sucked into jet engines. Assignments Today : P. 328 – 332 For Tomorrow : Exercises 8.18, 8.21–23, 8.25, 8.27 Monday : P. 334 – 338, 347–351, Tuesday : Exercises 8.29, 8.33, 8.34, 8.38–41, 8.59, 8.61, 8.67–69 217 STA 2023 c B.Presnell & D.Wackerly - Lecture 19 218 Last Time: ¤ £¡ ¢ to within “B” units with ¨¨¢ ©§¦ ¥ Estimate conﬁdence. and solve for . (p. 307) % \$ " !    # Parts of a statistical test. (p. 322) & The hypothesis of MAIN INTEREST is the ALTERNATIVE or RESEARCH hypothesis, – 1 0 ( , ¦642 5753¢ ' )¡ ( ' light bulb ex. . (p. 322) What we are “trying to prove” in an objective, fair manner The “other” hypothesis is called the NULL 8 ' & HYPOTHESIS, – , light bulb ex. (p. 322) ¦642 575©¢ # 0 8 ' )¡ Errors: p. 325 Reality Decision Ho true Ha true Accept Ho Correct Type II error Reject Ho Type I error Correct STA 2023 c B.Presnell & D.Wackerly - Lecture 19 (p. 323), SIGNIFICANCE ¦ Type I error 219 ¡ & # ¥ LEVEL of the test. (p. 325) ¦ Type II error ¡£¤ ¡ ¡ # ( ' ¥ & ¥ & 642 75©¢ ¥ # & ¥ The test that we will discuss have the SMALLEST ¥ for the ’s that we pick. ¡ when ¦642 §5©¢ 1 saying & ¥ In our lightbulb example, # & ¥ ¥ # ¡£¢ saying what we “want” to say when we should not ’s ¦ ' ¤ # true 8 when ¦ ¥ accepting ¢ and/or STA 2023 c B.Presnell & D.Wackerly - Lecture 19 220 Parts of a Statistical Test (p. 326) . ' ' 2. Alternative Hypothesis : ( 8 1. Null Hypothesis : . 3. Test Statistic : (TS) & computed from the sample data using a formula & forms the basis for our decision. 4. Rejection Region : (RR) & & & & Get data & Compute value of TS & Make decision ¥ Do experiment Then ' depends on the choice of 8 gives values of TS for which is REJECTED STA 2023 c B.Presnell & D.Wackerly - Lecture 19 221 Decision : & If the value of the TS is in the REJECTION ' and ( REGION, we . ' 8 If the value of the TS is NOT in the REJECTION REGION, we DO NOT REJECT. because we usually do 8 ' – We do not ACCEPT not know the probability of making an error if we do so. 8 ' – If we accept , what kind of error could we make? – What is the probability of a TYPE II error? usually ' judgement 8 Don’t want to accept , so we reserve ' that is really true ( depends on the value of the parameter in & STA 2023 c B.Presnell & D.Wackerly - Lecture 19 222 Courtroom Analogy ( & 8 : innocent : guilty ' & ' & Experimenter : Prosecutor & Put burden of proof on Prosecutor : Experimenter & Proof “beyond a reasonable doubt” : small. ¥ STA 2023 c B.Presnell & D.Wackerly - Lecture 19 223 Large Sample Hypothesis Testing 8 8 # 1 0 0 8 ( ' ' 1 0 0 8 ( ' ' #8 & & Rejection Region, RR 62 4 3¢ # Test Statistic, TS 62 4 3¢ Need . a ﬁxed particular value of Lightbulb Example : In this case, about a Population Mean, STA 2023 c B.Presnell & D.Wackerly - Lecture 19 224 ¦ 2 ¨ Recall : Large Sample %¡ & has an APPROXIMATE distribution ¡ £ ¢ % ¤ ¤ ¡ ¤ ¦ £ # # £ % ¢ # & & That is ¡ ¤ ¥ ¢ £ is a is true ¤ ¤ 0 8 ' & ¡ 8 # ¦ # ¤ ¡ 8 8 if ¤ ¥ £ % ¢  # £ is a TEST STATISTIC is true, ¤ 8 ¦ % ¤ ¡ ¤ £ ' # 8 ¡ ¥ ¤ ¢ £ #  has a 8 If distribution & STA 2023 c B.Presnell & D.Wackerly - Lecture 19 is close to the true value of ¡ FACT: 225 , whatever that true value is. is POSITIVE and LARGE 0 8 & ' in favor of 0 1 “something” ¥ # ( ¦  ' zα α 1 1 0 ( ' ¥ Rejection region : type I error & Rejection region : 0 # If we are interested in & Want than & Should is probably & ¡ The true value of by a “lot” of standard errors. than 8 is 8  If . & STA 2023 c B.Presnell & D.Wackerly - Lecture 19 226 ¤ ¡ ¤ £ ¦ % ¢ 8 # 8 1 0 8 0 ¤ ' ( ' ¡ ¤ ¦ £ %  #  1 ¡  0 0 ¤¤ “Upper Tail test”, “One Tail Test” (p. 329) Ex. : Lightbulb Example true mean lifelength of ALL BULBS # 642 75©¢ 1 642 75©¢ 0 8 0 # # 2 2 4 ¦ 4 £ 2 2 # ¨ ¦ ' ( ' ¡ 2 3¢ # & # ¦ §¥ # 6 5¨ % # & ¥ Data : level test, RR : #  & STA 2023 c B.Presnell & D.Wackerly - Lecture 19 227 & Conclusion : AT THE ( ' in favor of 8 ' LEVEL!! 6 5¨ # ¥ & In terms of this problem: 642 75©¢ ¦ conﬁdence ).  56 ¢¨ # ¥ If we wanted , is level of 6¨ signiﬁcance” ( or with at the “ Claim that the mean lifelength of all bulbs, – RR : 4 1 6 ¦ ¢ #  ¢¨ at the – 2 72 – Conclusion : Is ? level ' 8 – In terms of this problem claim that the mean lifelength of all 642 5©¢ conﬁdence ).  ¦¦ with at the bulbs is larger than ¢¨ “ – NOTE: This DOES NOT mean that is true!! level” ( or 642 75©¢ # 0 ' & STA 2023 c B.Presnell & D.Wackerly - Lecture 19 228 Ex. : #8.24, p. 333 Manufacturer inspection equip. for printed circuit boards claims that “product can inspect, on average, at least 10 boards per second”. Evidence to % 6¨ # ¥ . ? 0 8 ( ' and # one-second runs? Use ¥ “refute the claim” based on data for ' 0 8 ( & ' ' If we are interested in : ¤ ¦ % ¡ ¤ £ ¢ 0 ¡ 8 0 ¤ ' ( ' ¡ ¤ ¦ % 8 # 8 £ #  0 ¡ ¤¤ 0 α − Zα 0 “Lower Tail test”, (p. 329) STA 2023 c B.Presnell & D.Wackerly - Lecture 19 229 Back to #8.24: & Data: 10 9 10 10 11 9 12 8 7 10 11 9 9 13 9 10 9 9 9 7 12 6 9 10 11 12 10 0 10 11 12 9 8 9 6 10 11 10 12 8 10 8 7 9 7 9 9 10 2¨ 73¢ £ # 4 4 ¦ 4 ¦ ¡ # # £ level test 6 5¨ % # & ¥ & RR : ¨¤ 2 72  # # level of signiﬁcance. In terms of this 4¤ ¥ ¨ 2 6¨ application: “There # ¨ at the enough evidence at the level to indicate that the mean number of circuit 6 5¨ boards inspected per second is less than 10 .” & ...
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