Week7-2up - EXAMS RETURNED Exercises 5.24 5.25 5.31 5.36–37 5.39–40 For Thursday Read more on pages 198–208 For Wednesday Exercises 5.15 5.16

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Unformatted text preview: EXAMS RETURNED Exercises 5.24, 5.25, 5.31, 5.36–37, 5.39–40 For Thursday: Read more on pages 198–208 For Wednesday: Exercises 5.15, 5.16, 5.19, 5.20, For tomorrow: £ ¢ number of trials, ¨ £ © ¥ ¨   ¢ © Mean (p. 185) : § ¡ ¡ 0 1 2 Tables : Contain Variance (p. 185) : ¡   # Today : pages 210-11, 215–226 If  $ 3 ¤ © ¥ ¡ ¡ ¨ £ "¢ for ..... & % ¦¥ ¤  ' Assignments : k k+1 for ' § © £ ¢ " ¨ § : (p. 1183) ¨ £  " ! ..... £  Binomial Experiment p. 179 " n-1 ! because of our deep need to complain” - Lily Tomlin " ( £ Last Time : The Binomial Probability Distribution STA 2023 c B.Presnell & D.Wackerly - Lecture 11 n ( Thought: “I personally think we developed language 133  ( STA 2023 c B.Presnell & D.Wackerly - Lecture 11 ¢ ¢ " 134 135 20 times and record © number of correct Let – If the psychic is guessing, what is – © and and . . correct ? is a binomial random variable with ¢ ¦ – What is a “success” in this case? # correct choices in 20 trys correct § ¨ ¢ % $ ¡ ¡ £ ¥ ¤ © ¤ ¥ identifications. Assume 20 trials are indep. £ Psychic tries to identify card (without looking). Do this Ex. Claimed Psychic. Five cards shuffled, one chosen. STA 2023 c B.Presnell & D.Wackerly - Lecture 11 £ 0 2 3 ..... 5 6 ..... 19 Look for “jumps” between the possible values. Countable number of distinct values Discrete Random Variables (Table II(h), p. 772) 1 times”? What is prob. psychic is correct “6 or more STA 2023 c B.Presnell & D.Wackerly - Lecture 11 ¡ § value. Assign a “chunk” of probability to each distinct ¡ £ # # £ $ ¨ ¢ § ¡ £ £ £ £ £ ¨ ( ( ¤ © ¥ ¤ © ¥ ¤ ©¥ ¡ ¡ ¡ £ £ ¢ § ¢ § ¢ 20 136 ¡ ¡ ¡ ¡ histogram A mathematical model for the population (p. 210). – The “density” can be written a function of , – The depth or density depends on the value of . – It will pile up in certain places. interval like we would spread a handful of sand. Distribute our 1 unit of probability over appropriate to each Too many values go assign a “chunk” of probability etc. No gaps between values : Time, Height, Pressure, ¡ more line intervals graph of f(x) a P(a<X<b) Total Area = 1 is b (p. 210). Use a table. Calculus (not in this class) – How do I find these areas? a value in that interval (p. 210). corresponds to the probability that 138 will take Over a given interval, the area under the graph of graph of f(x) The total area under graph of Take on values associated with points on one or © ¥ § § § ¡ STA 2023 c B.Presnell & D.Wackerly - Lecture 11 ¡ 137 ¡ © ¥ Continuous Random Variables © © ¥ STA 2023 c B.Presnell & D.Wackerly - Lecture 11 ¡ © ! © ¤ ¥ is a continuous r.v., & © If a ¢ ¤ ©¥ (p. 210). probs. of intervals, unlike discrete ¡ © if its probability density is (p. 216) and std. dev. Recall:  § don’t make any difference in ¢ A random variable has a normal dist. with mean Will write with mean N and std dev ” and std dev ” MEANS “ has a normal distribution normal distribution with mean N 140 As a shorthand for “ has a Formula traces out the “bell shaped curve”. No need to know this formula. ¥ so, for contin. r.v.’s, “endpoints” & One very important continuous distribution. §  £ £ & ¤ £ § ¥ ¤ ¥ ¤ £  £ ©& ¢ © Line has area 0 ¡ The Normal Distribution  for all , £ § ¥ ¤ § ¡ ¥ § %  Note: ¢ STA 2023 c B.Presnell & D.Wackerly - Lecture 11 ¤ ¡ ¡ ¡ © §  £ ¥   ¤  © 139 ¤ "  © ! ! £ ¤ $  ¡  ¥ # ¡ " " © ¥  ! £ " § ¡  ! ¨ © " § " $ §  " ¥ ¦ # $ " © " $ STA 2023 c B.Presnell & D.Wackerly - Lecture 11 ¤ © " ¡ ¦ ! ! # 0 4 8 10 12 14 µ =12 σ =1 5 10 15 µ =10 σ =3 Two Normal Curves with Same Mean and Differnent Std Devs µ =10 σ =1 Two Normal Curves with Diff Means, Same Std Dev 6 µ =8 σ =1 20 16 is normal with mean If deviations . . . is larger than is from The -score gives the number of standard and std dev , is said to have a standard normal Recall : -score : i.e., distribution if . is smaller than That is, the value of is already a -score. . is a standard normal random variable: 5.6, p. 216) The r. v.  measures the spread of the distribution. (See Figure ¢ ¢ £ £ ,    © . The std dev, £ bell-shaped curve, centered at ¡ © £ © £ © § $ # The Standard Normal Distribution – p. 217 ¥  § # The graph of the normal density is a symmetric, © ¡ £ £ © # $ STA 2023 c B.Presnell & D.Wackerly - Lecture 11 ! 141  ¡ © £ "! © # § © £© ¡ # # # STA 2023 c B.Presnell & D.Wackerly - Lecture 11 ¡ ¡ ¡ § " $  ! 142 1/2 0 1/2 given in Table IV, p. 809 . curves between z-scores of  0 Find -z z Find 0 Tabled area and , for z Find Given in table are For any normal random variable, areas under normal  143 .3643 .3849 .4032 .4192 .4332 . . . .4713 .4772 0.0 0.1 0.2 . . . 0.9 1.0 1.1 1.2 1.3 1.4 1.5 . . . 1.9 2.0 2.1 2.2 . . . 2.9 3.0 .4821 .4861 . . . .4981 .4987 .00 .0000 .0398 .0793 . . . .3159 .3414 .4826 .4864 . . . .4982 .4987 .3665 .3869 .4049 .4207 .4345 . . . .4719 .4778 .01 .0040 .0438 .0832 . . . .3186 .3438 .4830 .4868 . . . .4982 .4987 .3686 .3888 .4066 .4222 4357 . . . .4726 .4783 .02 .0080 .0478 .0871 . . . .3212 .3461 ... ... ... ... .. . ... ... ... ... ... ... ... .. . ... ... ... ... ... ... .. . .4854 .4887 . . . .4986 .4990 .3810 .3997 .4162 .4306 .4429 . . . .4761 .4812 .08 .0319 .0714 .1103 . . . .3365 .3599 (Part of the) Standard Normal Table STA 2023 c B.Presnell & D.Wackerly - Lecture 11 STA 2023 c B.Presnell & D.Wackerly - Lecture 11 ¢ .4857 .4890 . . . .4986 .4990 .3830 .4015 .4177 .4319 .4441 . . . .4767 .4817 .09 .0359 .0753 .1141 . . . .3389 .3621 144 Find Find ¡ ¡ ¡ ¤ ¥ ¤ ¥ ¤ ¤ ¥ ¤ ¥  ¥  Find ¢ is a standard normal r.v., then ¡ " ¥ ¤ ¢ ¢ " ¢ " "! Ex. If   §  ! ! " ¡ ! ! "  § § ¢ " ¢ § " § § !  "  £ £ £ £ STA 2023 c B.Presnell & D.Wackerly - Lecture 11 £ 145 146 Exercises 6.1, 6.3, 6.4, 6.8 For Tuesday: Read pages 254–264, COMPUTER DEMO For Monday: 5.96, 5.98 Exercises 5.24, 5.25, 5.31, 5.36–37,5.39–40, 5.95, For tomorrow: Today : P. 215–226 Assignments : for you. Thought: If at first you don’t succeed, skydiving is not STA 2023 c B.Presnell & D.Wackerly - Lecture 12 ¤ & © ¡ § b and , for pictures Key to finding correct areas (probabilities) : draw in Table IV, p. 809 . Areas under normal curves between z-scores of ¢ ¡ £  Find Find ¤ ¤ § ¥ # ¡ ¡ $ ¤ ¥ , find: is a STANDARD NORMAL random variable ¢ ¤ ¥ Normal distribution is special case. is a continuous r.v., (p. 210) a ¢ If ¤ £ ©& ¢ © ¡ P(a<X<b) £ graph of f(x) ¡ ! If (p. 210). ¤ § ¡ & ¥ § ¤ ¥ ¤ ¤ £ § " ¥ § " Probabilities are areas under “density function” £ ¤ ¢ Last Time : Continuous Random Variables ¡ & ¢ ¡ ¢ © §  ¥  ¡ ¡  STA 2023 c B.Presnell & D.Wackerly - Lecture 12  ! ¤ ¥ ¤ 147 §  STA 2023 c B.Presnell & D.Wackerly - Lecture 12 ¤ ! ! & " & & § & § " ¥ ¤ £ " " § ¤ § £ !  § £ £ 148 ¤ ¤ ¢ ¢  such that  ¢  ¤ ¤   ¤ ¢ ¤ ¢ § ¤ § ¤ Find a number  ¥ ¤ ¤ ¥ ¡ ¡ £ £ ¥ ¤ & such that  ¤ ¥ Find §  ! "  ¢ " ¤  Note : Used table in reverse. ¢ & ¢ " ! "  Find a number ¡ ¡ have a standard normal dist. ¢ § ¤ Ex. Let ¥ ¤ § § ¥ ¤ & STA 2023 c B.Presnell & D.Wackerly - Lecture 12 § ¤  ¤ Recall the Empirical Rule!!! 149 £ "£ § & & Find  "£ "£ £ ¥  £  "£ "£ § § ¥    STA 2023 c B.Presnell & D.Wackerly - Lecture 12  . . 150  ¤ © ¥ ¢ score : § #  § £   $    © ¢  ¡ £ " £ . score , its score is    § © Want  " Employer requires dexterity score of at least 80.  ¡ ¡ §  ¢ § ¤ © ¥ ¤ ©¥ © £ § £ £ £ £ £  ©  – Whatever the value of Normal distibution,  so that "! – Find  score on dexterity test  Want - 98th percentile of dexterity scores.  Ex.: #5.31, p. 228 ¡ STA 2023 c B.Presnell & D.Wackerly - Lecture 12 £ £ £ 151  " STA 2023 c B.Presnell & D.Wackerly - Lecture 12  152 ! £ $  #  ¢  score : ¤ § . What fraction use opener beyond the limit? ! £ © Limit for interference: £ " Normally distributed, 153 probability that all 5 used the openers beyond 50’? Five users randomly selected. What is the ¤ © ¥ § ¥ 154 Find to so that only 2% of cans are so that amount dispensed. unusable. What should we set More than 5 ml dispensed – paint unusable. something we can set paint is normally distributed, numbers) Amount of dye, in ml., dispensed into can of Ex. : Like #5.40, p. 229 (Same scenario, different – ? do you think about the claim that openers and all 5 use the opener beyond 50’, what If you randomly select 5 owners of garage door STA 2023 c B.Presnell & D.Wackerly - Lecture 12  distance at which garage door opener is used.  $ Ex. Not in book ¢ " ¤ "   § " ! # ¤ © ¥ $ ! ¢ "£ ¡ ¡  STA 2023 c B.Presnell & D.Wackerly - Lecture 12 ¡ § # © ¡  ¡ £ # £ £ ! £  £ # " # . ¢ ¤ score for " 0 .48 score for 5 is must be approx. z0 §  ¡ ¡ § ¡  Thus, the ¤ From table,  £  # § £ .5 # ¤ § £  "  , the  Whatever the value of  ¥ " § STA 2023 c B.Presnell & D.Wackerly - Lecture 12  milliliters   £ £ £ £ " § ! ¤ ¥ "  ¥ " §  155 ...
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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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