Week6 - STA 2023 c B.Presnell& D.Wackerly Lecture 10...

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