Week5-2up - Exam 1 During your discussion session For...

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Unformatted text preview: Exam 1 During your discussion session For Tuesday 10/1/02: exam, problems, course content Question and Answers ask questions about For Monday 9/30/02: Exercises 4.33, 4.36, 4.37, 4.50(b), 4.51 ¢ £  § ¨ §  ¢ ¤ £  £ ¤ £ ¤ ¢    ¦ ¦ ¥ ¦  ¦ £ ¦ §  § ¦ ¢ £ ¨ § ¢ ¨  ¦ § § £ ¤ ¥ ¦ £ ¤ £ ¤  §  ¢  ¦ §  § §  ¤ ¥ ¢ ¢ ¦ £ ¤ § ¢ (F. S.) (p. 169) Probability Distribution for Discrete Ran. Var. (p. 166). Discrete and Continuous Random Variables  £ £ ¤ £ ¤ Put the pieces together!  (F. S.) ¥ For Thursday 9/26/02:  mutually exclusive  For Wednesday : Pages 179 – 185 ¢ ¦ Exercises 4.22, 4.27-29 ¢ For Tuesday: § (F. S.)  © . What is ¢ For Today: pages 172–176 ¦ § © ¦ Altzheimers Example (Like #3.109, p. 158)  £ ¤ £ ¤ ¢ ¢ £ ¤ © and § Assignments : ¤ Know ¨ can count; and those who can’t. ¨ © ¢ £ ¤ ¤ Last Time : ¥ ¦ © © © ¥ ¦ STA 2023 c D.Wackerly - Lecture 8 § Thought: There are three kinds of people: those who 101 ¡ ¡ ¡ ¨ STA 2023 c D.Wackerly - Lecture 8 § ? 102 TTT TTH THT THH HTT HTH HHT HHH Probability ¢£ Point § # matches won £ §  § about So, average value of times times; over 600 repetitions should be roughly roughly large number of times, say 600. Expect Consider the lightbulb example. Repeat experiment a The Expected Value a Random Variable Sample £ by H. £ ¡ £ ¢£ £ ¡ £ ¡ £ ¡ £ ¦ ¢£ ¢£ ¢£ ¢£ § § §     ¢£ ¢£ ¢£ ¡ £ ¦ £ ¦ § § §     ¡ £ ¦ £ ¡ £ ¦ ¢£ ¢£ ¢£ § § §    ¡ ¡ ¡ ¦  ¢¡ ¤ ¢¡ § ¢¡ § ¢¡ ¢¡ § ¢¡ ¢¡ ¢¡ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ©  ¡    ¨ ¡ ¦ ¦ ¨     § will win. H and T play 3 matches. § § Ex. Raquetball: odds are 2:1 that when H and T play, H    ¨    ¨  ¨ ¨ £  ¡  ¦ £  ¨ ¨   ¨  ¨  ¦   £     ¨ §      ¦ ¦  ¨ ¡  ¨ ¨ § ¨ © ¨ ¨ £ § STA 2023 c D.Wackerly - Lecture 8  103  STA 2023 c D.Wackerly - Lecture 8     § ¦ ¨     ¡  ¦ ¨ ¨ ¨ £  ¨ £ ¦ © £ ¨ § ¨ £   ¨ ¨ § ¦  ¡  £ 104 of . (population mean of ) is § © £  § ¡ £  is NOT necessarily a possible value of . is a “weighted average” of the possible values period has distribution EXAMPLE : Suppose # of calls to an office in a 5 min. STA 2023 c D.Wackerly - Lecture 8 § ¡ £ defined to be § £ ¨ ¦ ¡ £ expected value of § © ¦ £   ¦   Def. (Def. 4.5, p. 172) For a discrete r.v. , the £ £ ¡ ¡ ¡ ¡ 105 § STA 2023 c D.Wackerly - Lecture 8     ¢ 106 . § © £    ¨ § £ ¡ ¢ £  § £ £ §  ¡ £ ¡ The standard deviation of . £ is (Def. 4.7, p. 174) is ¢  A shortcut formula for the variance of § Recall that  profit. © §     rain  No rain, profit of $20,000, Rain, lose $12,000 ¡ ¢    ¤ is £ § EXAMPLE : #4.27 p. 177 Rock concert: §  ¥ £  §     § 108 Def. (Def. 4.6, p. 174) The variance of a discrete r.v., , £ STA 2023 c D.Wackerly - Lecture 8 ¢ ¢ © 107 § ¦ ¨   © ¢    STA 2023 c D.Wackerly - Lecture 8  ¡ £ ¡  ¦ 109 ¢   and is . – The “approximately” – Empirical Rule – The “at least” – Tchebysheff. better” than If we have the probability distribution, we can “do Empirical Rule (if mound shaped) Tchebysheff’s Theorem (always applies) The population mean and standard deviation Interpretation of ¢ § ¨ ¦ ¡ £ © £ ¦ £   The standard deviation of ¦   STA 2023 c D.Wackerly - Lecture 8  EXAMPLE : Telephone Calls § STA 2023 c D.Wackerly - Lecture 8      ¡ ¡ ¡  £ ¡ 110  § £  ¢ ¢ ¤ § ¡ £    £ ¡ ¡ ¢  © © © ¤  ¤ ¡ ¨ ¡ ¦ and ¢ ¡ £ Estimate prob. of fatal crash to be ¢  £ ¨ ¦ ¢ ¦ ¢ ¡ from a single policy. ¨    £ ¤ between . ¨ ¦  . be the “profit” to the insurance company § crash. Let ¨ ¡ § §   ¡ 112 costing $10 will pay $1 million if the passenger dies in a £  and 285 fatalities due to air accidents. Flight insurance ¨ £ £ 455 million revenue air passengers 1988, there were  ¢ § ¡ ¨ ¨ ¨  are   ¤  ¨ Ex. According to 1990 Statistical Abstract of the U.S., of ¨    ¡  ¢ © between  § § STA 2023 c D.Wackerly - Lecture 8 § Possible values of   ¨ EXAMPLE : Telephone calls 111 ¨ STA 2023 c D.Wackerly - Lecture 8 ¨ ¡  ¨ , resp. daily sales. Sale(s) made. How many contacts were made. A sample point must indicate: Find the distribution of $50,000 sale – probability is no sale – probability is Each contact results in and ¡ ¦ ¦ ¡ ¢ day. Probabilities are ¡   Probability Sales 2 contacts, sale on 1st, no on 2nd 1 contact, sale Sample Point ¢   ¡          STA 2023 c D.Wackerly - Lecture 8  ¡ ¡ ¡ ¡ ¦ ¡ ¡ ¡ ¡ ¡ ¢ ¢ ¢ ¢                  ¡ ¡ ¡  ©         £     EXAMPLE : Salesman contacts 1 or 2 customers per 113 ¦ STA 2023 c D.Wackerly - Lecture 8 § ¢ ¨ ¨ ¨ ¨ £  ¤  £   ¨ ¨ ¢ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¢ ¨ ¨ ¨ ¨ 114    £ ¢ ¢ ¢ ¤ ¡ ¡ £ ¤ ¡         ¡ © £  ¢ £ 115 What is the probability that STA 2023 c D.Wackerly - Lecture 8 ¨ £ §  ¤£ ¢ £  ¨    © ¢ ¨  § £ ¢   ¦  £ ¢ £ ¦  ¦ £ ¦ ¢ £ § ¦ ¦ ¦ £ ¥ §  £ §  £ § £ ¨ ¦ ¦ ¤ ¨ § ¥ Thus, for the sales example,  ¨ ¤     ¢ ¨ ¤ ¦ ¢ ¨  ¥  ¡    ¤ ¥   ¦  ¤£ ¦§    ¡ ¡ STA 2023 c D.Wackerly - Lecture 8 is in this interval ? 116 4.93, 4.95, 4.97, 4.99, 4.101, PROJECT 2. For Thursday : Exercises 4.38, 4.45, 4.47, 4.49-52, For Wednesday : Pages 185 – 189, MINITAB Exam 1 During your discussion session For Tuesday 10/1/02: exam, problems, course content Question and Answers ask questions about  § ¡ £  ¡ £ ¡  § £ £ £  and , resp. Find the distribution of daily sales. $50,000 sale – probability is no sale – probability is Each contact results in day. Probabilities are EXAMPLE : Salesman contacts 1 or 2 customers per (p. 174) is § The standard deviation of © For Monday 9/30/02: Variance p. 174 ¡ ¡ ¡ ¡ Exercises 4.33, 4.36, 4.37, 4.50(b), 4.51 ¢   For Thursday :    ¡  Today : Pages 179 – 185 § Mean p. 172  £  © change in the basement. ¢ Last Time : § STA 2023 c D.Wackerly - Lecture 9  © Thought: People who live in glass houses should 117  STA 2023 c D.Wackerly - Lecture 9   118 .   ¨ ¢  £       ¤  ¨ ¡ £ £  £   ¨ £ ¡ £  ¡ ¥  £ ¡  § ¡ ¨   £ ¨ ¢  £ ¨ ¢ ¢   ¦  § £ ¥ ¨ ¢  £ © ¡ ¨ £ ¤  ¨ ¡ £  ¢ ¦ £ £ § ¨ ¢ £ ¨ ¨ ¨  ¢ £   ¨  £ ¨ ¨ ¨ ¢ ¨  ¨ ¨ ¨ £ ¨  § ¥ ¨  ¨ ¨     §    ¨   ¤ ¤  ¨  ¨    ¤   ©   £  § ¦  ¨ ¨ ¦   STA 2023 c D.Wackerly - Lecture 9  £ ¤ ¨ £ ¤£ ¢ ¢  ¦ £ ¦ £ § § ¨ £ £ £ ¨ ¤  £  § ¨  ¨   ¨   ¨ ¨ ¨  ¨  ¨ ¨ ¥ £ ¨ £  ¡  ¨  119 ¨ £ ¡ ¨ ¨ £ – is the value 0. is in interval that is in the interval – The only value of What is the probability that ¥ (with positive probability) ¢ §  £ © ¤ ¡ ¨ ¢   ¤ £ ¦ ¢ ¢ £  ¢ £ ¦ ¦ £ ¦ ¦ £  § ¦ £  ¥ ¨  ¦ ¦ ¡ ¨ ¢ is in this interval ? Thus, for the sales example, £ ¦ § ¨ £ ¡  ¨     STA 2023 c D.Wackerly - Lecture 9 ¡ ¡ ¢ £ ¢ ¦ ¦ §   ¢ ¥  ¤ ¦§ § ¦      ¡ ¨ ¢  ¥ ¨ ¢ ¥  ¤£ ¤ ¥  ¦§ ¤£ ¦    § 120 3 coins 7 guesses 4 golfers 3 patients 5 insects 4 patients 3 coins 3 games 3.47 3.53 3.55 3.56 3.105 3.107 4.11 Example Any similarities? 3 coins # H wins # heads # alive # attracted # recover # h-in-1 # correct HTH HTT AADA AANNA SSF SSFF YYYNNYN HTT £  ¡ ¡ , , . . ’s in the Ex. Choose 10 microwave ovens at random from # H in 10 tosses, Ex. Toss a fair coin 10 times, trials. 5. Variable of interest is , the number of 4. Trials are independent. # heads stays same from trial to trial. ¢ ¡ 3. Prob. of § HTT ¡ # heads =good). unknown. factory output. Count number defective ( =defective, ¡ 3.21 ¢ .  or £ A sample point ¢ Of interest © © Experiment identical trials in experiment. ¡ Problem   2. Each trial results in one of two possible outcomes, 1. ¡ class examples? © £ A “pattern” emerging in (some) homework problems, © ¡ ¢ Binomial Experiment p. 179 STA 2023 c D.Wackerly - Lecture 9 ¡ ¦  ¦  ¨ ¤ ¢¡  ¦ § 121  ¡ STA 2023 c D.Wackerly - Lecture 9 122 §  £ ¢ , read £ £ if . choose . ¢  size size committee of . EX. : Club consists of 10 members, select a if is read § ¡ objects , where the order in which the ¡ is a positive integer, ¢ factorial is © © ¦ Def. : (p. 139) If £ £ £ ¢ ¢ ¢ ¡ ¡ ¡ © © © © © Some Preliminary Results  ¥   ¡ objects are selected does not matter is ¡ ¡ Variable, . –The Probability distribution of . ¡ ¡ from among  associated with all values of a Binomial Random ¡ : ¨ Objective : Obtain a formula for the probabilities Fact: The number of ways of selecting £ ¡ STA 2023 c D.Wackerly - Lecture 9 £ Ex. Political Poll : Likes Bush, Not 123  Ex. Diagnostic device : detects or misses ¢ ¡ =Thomas wins § ¡ ¨ ¡ ¡ ¡ ¡  ¡ ¡ =Henry wins, ¤ Ex. Play three games of racquetball  ¨    ¦   STA 2023 c D.Wackerly - Lecture 9   ¦ ¢ ¦ £ ¦  ¢ ¦ ¦     ¨ ¡   124 ©  ¨ ¢ © £ ¢  ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ § ¡ § Sample Points  .       . Probabilities – toss 3 coins, H and T play 3 games, etc. ¡  £ ¢ © Ex. : ¢ £ Try the formula Thus probability Same as sample point approach!! Thus, . Each single sample point where . distinct sample points corresponding to There are Note £ ¦ for ¨  ¡ ¡ ¡ ¡ ¥ ¦ ¦ Why this formula? Sample point approach: ¡ © ¡ £ are  Possible values of § trials and £ ©  . (p. 183) ¡ ¡ ¡ ¡ ¡  is a binomial random variable with  If ¢ STA 2023 c D.Wackerly - Lecture 9  ¡  © © © ¢ ¢ ¢  125 STA 2023 c D.Wackerly - Lecture 9 ¢ § ¡ ¡  ©      © © ¡ ¢ £ ¡ § £ ¡ © ¡ © © © © £ ¡ © ¦ ¦ ¦ ¨  ¡  has . 126  ©  £ £ ¡    § £ ¡ £ ¢£ £  ¢£ There are . Each has probability  ¢£ ¢£ £ ¡ £ ¡ § § ¡ £ ¡ £ ¡ ¦ £ ¡ ¢£ ¢£ ¢¡ ¢¡ ¥ ¥ ¥ § § § # won by H. © ,   ¡  ¡ ¡ # with errors ¢ ¤ ¦ (i) exactly 2 sheets contain errors. £ § ¢¡ ¢¡ £ ¡ 2 1 2 1 1 0 § § § sample points with , ¢ £ ¡ ¢£ £  § ¢¡ ¢¡ ¢¡ Find the probability that: § ¥ ¥ ¥ contain errors. 5 randomly chosen sheets examined.  3 £     © £ ¢£ ¢£ ¡  £ ¡ © HHH HHT HTH HTT THH THT TTH TTT ¡ Ex. Accountant believes 10% of all ledger sheets £ Probability  Outcome £ ¦ § § 2/3 of winning any match. Matches indep. ¢ Ex. H and T play 3 matches of racquetball. H has prob. ¢£   ¨  ¨¦      § ¡¥ ¨ ¨ ¡¥  ¨ § ¡¥  ¥ STA 2023 c D.Wackerly - Lecture 9 § 127 STA 2023 c D.Wackerly - Lecture 9   128  § §   ¢ £ § © ¢ ¢ £ £ £  £ ¢  £ § §  £ £ ¤¦   ¦  § £  ¨  § £ £  §  §   §  ¦ £   £ ¢ so, ¨ ¢ ¤ ¡  ¢ ¢ £ £  £ more work  £ ¦ © £ ¨ ¢ ¢ £ ¢  ¢  § £ £ £ (iii) more than 2 contain errors. ¢ ¢    and § from (i), and ¡    ¢ ¡  £  Note: The following require no calculation: ¦   ¦  § ¡¥ ¨ ¢ ¡ £ © ¡ ¥  £ ¦ ¢  §  (v) more than 1 but no more than 3 contain errors.  ¦ § © ¨ ¦  £ ¡ ¦  § ¦ ¢ ¦ § (iv) at least one contains errors. § ¦ £ £ ¤ (ii) at most 2 contain errors. § STA 2023 c D.Wackerly - Lecture 9 ¡ § 129   £ § ¡¥ ©  ¡ ¥ ¢ ¦ £ ¤     £ § STA 2023 c D.Wackerly - Lecture 9  130 you expect?  ¢ ¢ ¢ ¢ If I toss a balanced coin 10 times, how many heads do ¡ . If What is the variance?  as a binomial random variable.  # who think statistics is an interesting subject   students). Can treat  § Here Ex. Sample of 20 students at UF (more than 37,000 £ ? switches, 132 . . What is the expected number of defective switches? © TREAT AS BINOMIAL If it is: § NOT BINOMIAL defectives. Is  © is discrete © found ¢ © is discrete £ switches are defective. Sample of ¢ =#defective  =#defective Ex. #4.51, p. 192 Supplier claims no more than  Choose 2 Choose 2  300 defective § 3 defective  700 good  and , sometimes , then (p. 185)  7 good bin  abbreviated 1000 items 10 items § is binomial with parameters ¢ If  # defective in sample.  Ex. Select sample of items from a shipment, and count STA 2023 c D.Wackerly - Lecture 9  131 ¢ STA 2023 c D.Wackerly - Lecture 9 ¨ ¨ ¢ © © © ¨  ¨  © © ©   ¨¦ ¨¦  ¥ © ©  ¥ ¨ ¡ ¡ ¨ ¢ ¦  © ¨ ¡  is a ¢ ¡ ’s) decimal places) binomial probs. ( for some © and ’s than in back of book). Use Minitab to compute binomial probs (more ’s ’s and Use tables in back of book to compute (correct to 3 ¡ Next Wednesday: binomial probabilities is easy but may be tedious. requirements for a binomial experiment), computing with a binomial random variable (checked all © ¡ 133 if the claim score indicates Observation: Once we recognize that we are dealing that – (Accept, Reject) the claim– – z-score for 4 is is correct? Is it likely that you would observe § § STA 2023 c D.Wackerly - Lecture 9 ...
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This note was uploaded on 12/15/2011 for the course STA 2023 taught by Professor Ripol during the Spring '08 term at University of Florida.

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