Week6-4up - 121 STA 2023 c B.Presnell & D.Wackerly...

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Unformatted text preview: 121 STA 2023 c B.Presnell & D.Wackerly - Lecture 10 STA 2023 c B.Presnell & D.Wackerly - Lecture 10 122 Binomial Experiment p. 179 1. change in the basement. ¡ Thought: People who live in glass houses should identical trials in experiment. 2. Each trial results in one of two possible outcomes, ¢ or £ . ¥ ¢ ©§¨¦¤ ¢ 3. Prob. of Today : Pages 185 – 189 For tomorrow: stays same from trial to trial. ©§ ¥¤  £ ¨¦¥  Assignments 4. Trials are independent. For Monday: Pages 210–211, 215–226 is the number of ’s in the 6. Binomial Distribution $ % & 1) '¤ ¡ #¥  !¤ © 0( & " “successes”: X UTS R V£ 96F 9P VWV£ I R EH69GP EE¢ I Q £ F F Q £ £ £ ¢ ¢ F F F9¢ ¢ ¢ positions Ex. Accountant believes 10% of all ledger sheets contain errors. 5 randomly chosen sheets examined. X UTS R 9P I R bP ¤ Y I Q Q FF c96F Y  Y  Y ¤ 69F `a`¤ FF Y (i) exactly 2 sheets contain errors. “successes”? , and then filling in the rest with ways to do this, so there are ¥ f ge sample points with exactly D 9 ¢ £ – There are positions to be ¥ ’s. D 9 filled with an of the C 7B – Think of selecting ¥ How many sample points with exactly “successes”. ¥  d 7yv2p© Bw2 8 w2 8  d B7€yv2u© B8 8  d 7xwu©  5v2u© . “ ’s” D 9¥ d D 9¥ “ ’s” : ¤ 9 D and “successes” is the product of $ A to # with errors "#¥  tB©!¤ ¥¡ the probability of any sample point corresponding ¥ s¤ terms ¥ A . Find the probability that: ¥ D 9¥ The probability of this sample point is terms So, ¥¢ A positions 124 $ $ ¥ 0( & D© ¤ © " r& 1) q¤ ¡ " ¥  9pih¥  !¤ CB 7¨¥ ¡ to . One sample point corresponding STA 2023 c B.Presnell & D.Wackerly - Lecture 10 A Suppose 123 STA 2023 c B.Presnell & D.Wackerly - Lecture 10 for trials. ¡ @9697653¥ 4888442 5. ¢ 4.95, 4.97, 4.99, 4.101 ¡ Exercises 4.37, 4.38, 4.45, 4.47, 4.49-52, 4.93, f ge A 125 STA 2023 c B.Presnell & D.Wackerly - Lecture 10 (iv) at least one contains errors. ©  § Dd 9 yd 8 ¥ d yv28 '©B vC8 Bw ¡ ¨ d 82 d8 8 ¢ © ¦§ 8u© ¤¥  u©'6 ¡ T  u©   p©  3¡ $ B d ¦8¤8 ¥ u© ¥  u© ¢ " d 8 8 $ ¢ T  p© S   u© £" ¡ (v) more than 1 but no more than 3 contain errors.  © p§ ¥  C i¤ © d 7 82 ¥  tBi¤ Bw © ¥ $   ¡ #¥ ¡ ¥ $  " ¡" ¡ ¥ ¥ so, more work  © p§ 127 ¥ C ¥ B © § ¥ ¡ ¡ ¥ ¢ 2 d ¢ ¥ 8 $¡ $2 ¡" ¥  ¥ ¡" (iii) more than 2 contain errors. © ¥  B  § © ¥  ¢ !¤ ¡  D !¤ ¡  C ih¥  B  § © © ©¤ STA 2023 c B.Presnell & D.Wackerly - Lecture 10 ¥ C from (i), and Note: The following require no calculation: and 126 ¥  (ii) at most 2 contain errors. STA 2023 c B.Presnell & D.Wackerly - Lecture 10 STA 2023 c B.Presnell & D.Wackerly - Lecture 10 128 Ex. Select sample of items from a shipment, and count Ex. #4.51, p. 192 Supplier claims no more than 2 82 ¥ 5yv213¦¤ 2 y2 ¢ ¥ ¡ switches are defective. Sample of TREAT AS BINOMIAL A What is the expected number of defective switches? you expect? . If . What is the variance? $ ¥ If I toss a balanced coin 10 times, how many heads do A as a binomial random variable. Here $ students). Can treat $ Ex. Sample of 20 students at UF (more than 37,000 # who think statistics is an interesting subject switches, ? d w 2 xw8 ¥ ¢ yd D 8 NOT BINOMIAL If it is: $ is discrete defectives. Is ¥ ¤¡ ¥ ¥ ¤¡ ¥ ¤ is discrete found ¥ ¦¤ ¡ ¥  G&¥ ©% 2 82 ¥ 5yv213¦¤ ¥¡ =#defective D¥ =#defective " # 8 Choose 2  Choose 2 ¤¡ ¥ 300 defective , then (p. 185) !` ¥ 3 defective bin and , sometimes ¤¡ 700 good   7 good ¤¡ ¥ ¤  4¤ ¡ © 1000 items ¡ abbreviated 10 items ¤ is binomial with parameters If # defective in sample. . 129 score indicates that   is a “rare event”. 130 A Portion of Table II (p. 771) D – Reject the claim– STA 2023 c B.Presnell & D.Wackerly - Lecture 10 if the claim  ' 8 ¢ d 5D ¥ ¤£¨¢§0¦¡¥ ¤¥¡ T ¥ 8 A – z-score for 4 is   is correct? D Is it likely that you would observe STA 2023 c B.Presnell & D.Wackerly - Lecture 10 n=10 p k 0.01 ... 0.60 0.70 ... 0.99 0 .904 ... .000 .000 ... .000 The table of the binomial distribution (pp. 770–773) 1 .996 ... .002 .000 ... .000 gives cumulative binomial probabilities, i.e., it gives 2 1.000 ... .012 .002 ... .000 3 1.000 ... .055 .011 ... .000 4 1.000 ... .166 .047 ... .000 5 1.000 ... .367 .150 ... .000 6 1.000 ... .618 .350 ... .000 7 1.000 ... .833 .617 ... .000 8 1.000 ... .954 .851 ... .004 9 1.000 ... .994 .972 ... .096 . Any needed prob. can  ¦ & ©  i¤  ¥  © FF © ©  © !¤ ¡ 96F ¡   !¤ ¡  2 !¤ ¥  k k+1 n-1 n : 2 3 4 5 6 (c) _ X>8 7 8 9 0 10 1 ©  § 3 4 5 6  w D § © ©§  h¥  ¨ ¥  ¥ ©§  h¥  ¨  [from Table II, p. 913] ©  § . _ X<7 7 8 9 10 __ 5<X<8 _ X<4 ©  w !¤ Ex. Conduct a survey, randomly select 10 indiv.: _ X<6 4 5 6 7 8 9 10 Interested in the number who are Lutherans ( X=7 ¥  w ¥ § © A ¥ [from Table II] Lutheran, any other (or no) rel. affil.) Interested in the number who are gun owners and favor gun control. ( ¥¢ 3 gun owner who favors gun control, everyone else) ¥£ 2 ¥£ 1 A 0 ¥¢ ¥ (b) _ X<8 2 _ X<7 132 “more than 4 and no more than 8.” ¥ ¥ ¥ ©§ ¨¥  ¨ 1 to find: STA 2023 c B.Presnell & D.Wackerly - Lecture 10 ¢ ¨ 0   (a) bin 842  xw616p© Ex. Use the table for 131 © § STA 2023 c B.Presnell & D.Wackerly - Lecture 10 3 D § © 2 ©  1 ¨ © § 0 .....  ©  .....  be obtained from these. ¨ © ¡ @996761¨¥ 4888442 for ...
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