Week7-4up - 133 STA 2023 c B.Presnell & D.Wackerly...

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Unformatted text preview: 133 STA 2023 c B.Presnell & D.Wackerly - Lecture 11 STA 2023 c B.Presnell & D.Wackerly - Lecture 11 Last Time : The Binomial Probability Distribution Thought: “I personally think we developed language because of our deep need to complain” - Lily Tomlin ¡ Binomial Experiment p. 179 ¡ 7X¢W98V586531US "TRP¥ 2V V2420 £ S Q ¦ 7#¢ £ F IH3CGE 7¢ £ D3CBA ¢@978578645301) ! (&$"  ¢  2722 £ '%# ! £ ¦  ¤¢  £ ¦ ©¨§¥ £ If Assignments : number of trials, : (p. 1183) for Today : pages 210-11, 215–226 ¡ EXAMS RETURNED Mean (p. 185) : ¡ Exercises 5.15, 5.16, 5.19, 5.20, For Wednesday: Variance (p. 185) : ¡ For tomorrow: 134 Read more on pages 198–208 Tables : Contain for ..... For Thursday: 0 1 2 3 ..... k k+1 n-1 n Exercises 5.24, 5.25, 5.31, 5.36–37, 5.39–40 STA 2023 c B.Presnell & D.Wackerly - Lecture 11 Ex. Claimed Psychic. Five cards shuffled, one chosen. Psychic tries to identify card (without looking). Do this £ ) 20 times and record What is prob. psychic is correct “6 or more times”? number of correct identifications. Assume 20 trials are indep. correct ? £  # correct choices in 20 trys 2 3 ..... 5 6 19 (Table II(h), p. 772) £) ¦ §¥ – What is a “success” in this case? Discrete Random Variables is a binomial random variable with ¡ £ RE Y £ BA Y £ 3¢CBA Y £ ¤¢  £ `¨ Y and Countable number of distinct values . Look for “jumps” between the possible values. . ¡ #b3C£ F E ¢ £ a and ¡ – 1 £XIeRPgV d c ¦ ¥ £d c ¦ ¥ XIeRPgV Let correct ¦ §¥ ¡ – ..... 0  If the psychic is guessing, what is 136 d c ¦ feR§¥ Y 135 STA 2023 c B.Presnell & D.Wackerly - Lecture 11 Assign a “chunk” of probability to each distinct value. 20 ¡ 137 STA 2023 c B.Presnell & D.Wackerly - Lecture 11 STA 2023 c B.Presnell & D.Wackerly - Lecture 11 138 graph of f(x) Continuous Random Variables ¡ Take on values associated with points on one or ¡ No gaps between values : Time, Height, Pressure, ¡ Too many values go assign a “chunk” of probability ¡ Distribute our 1 unit of probability over appropriate P(a<X<b) more line intervals a b 4  ¦ ¡ The total area under graph of etc. is graph of f(x) (p. 210). to each Total Area = 1 interval like we would spread a handful of sand. – It will pile up in certain places. ¡ Over a given interval, the area under the graph of  a value in that interval (p. 210). Calculus (not in this class) Y histogram Y 139 Use a table. ¡¡ ¢¢¡  ¦ A mathematical model for the population STA 2023 c B.Presnell & D.Wackerly - Lecture 11 will take – How do I find these areas? Y  (p. 210).  ¦ – The “density” can be written a function of , corresponds to the probability that  – The depth or density depends on the value of . STA 2023 c B.Presnell & D.Wackerly - Lecture 11 140 The Normal Distribution ¡ A random variable has a normal dist. with mean  E "  #F   !' ! ¦  4 £¦ E probs. of intervals, unlike discrete is a continuous r.v., 77 0(4& 7$ £ 887 1)"'4 (%¤ ¡  8¦§¨ ©¥£§¥ 8§ a¥£P¥ £ ¨ ¦ £ ¦ ¨ Q ¦ ¦ Q ¨ ¦ £ ¦ Q Q ¦ 8¤R ©¥£P¥ X8§e a¥£P¥ No need to know this formula. ¡ Formula traces out the “bell shaped curve”. As a shorthand for “ has a and std dev ” MEANS “ has a normal distribution  and std dev 74  9 74 A2 @(¦ ( 8 9 with mean  N A ¡ normal distribution with mean E 8 9 N  E gA¦ 2 Will write  ¡  If Recall: a  (p. 210). 554  176(437 2£ 0 ££ £ ¦ 13¤)§¥ don’t make any difference in if its probability density is (p. 216) 77 857 and std. dev. Line has area 0 £ so, for contin. r.v.’s, “endpoints” ¡ for all , A Note: One very important continuous distribution. ” 141 The r. v. is said to have a standard normal 5.6, p. 216) distribution if ¡ E  £ A   £ ¢¡  A£ ¥0 T) ¦1£ ¢¡  ¦e¨ ¢¡ ¥0  ¦0 ¤¢¡ ¥ £  ¡ £ 7 94 2( ¦0 ¡ 8 is normal with mean i.e., A 7AE  µ =12 σ =1 Recall : -score : 0 A measures the spread of the distribution. (See Figure µ =8 σ =1 142 4 E , . The std dev, STA 2023 c B.Presnell & D.Wackerly - Lecture 11 The Standard Normal Distribution – p. 217 The graph of the normal density is a symmetric, bell-shaped curve, centered at STA 2023 c B.Presnell & D.Wackerly - Lecture 11 and std dev , The -score gives the number of standard 14 deviations 16 Two Normal Curves with Diff Means, Same Std Dev µ =10 σ =1 is smaller than µ =10 σ =3 5 10 . is larger than If 0 is from 15 20 Two Normal Curves with Same Mean and Differnent Std Devs STA 2023 c B.Presnell & D.Wackerly - Lecture 11 . . is a standard normal random variable: That is, the value of 143 . . is already a -score. 12 10   T£ 0 4  8 A 6 A STA 2023 c B.Presnell & D.Wackerly - Lecture 11 144 (Part of the) Standard Normal Table For any normal random variable, areas under normal 0 given in Table IV, p. 809 . and , for 0 ¤ £ curves between z-scores of are Tabled area 0 z Given in table 1/2 1/2 -z 0 Find 0 Find z Find § 4 0.0 0.1 0.2 . . . 0.9 1.0 .00 .0000 .0398 .0793 . . . .3159 .3414 .01 .0040 .0438 .0832 . . . .3186 .3438 .02 .0080 .0478 .0871 . . . .3212 .3461 1.1 1.2 1.3 1.4 1.5 . . . 1.9 2.0 .3643 .3849 .4032 .4192 .4332 . . . .4713 .4772 .3665 .3869 .4049 .4207 .4345 . . . .4719 .4778 .3686 .3888 .4066 .4222 4357 . . . .4726 .4783 2.1 2.2 . . . 2.9 3.0 .4821 .4861 . . . .4981 .4987 .4826 .4864 . . . .4982 .4987 .4830 .4868 . . . .4982 .4987 ... ... ... ... .. . ... ... ... ... ... ... ... .. . ... ... ... ... .. . ... ... .08 .0319 .0714 .1103 . . . .3365 .3599 .09 .0359 .0753 .1141 . . . .3389 .3621 .3810 .3997 .4162 .4306 .4429 . . . .4761 .4812 .3830 .4015 .4177 .4319 .4441 . . . .4767 .4817 .4854 .4887 . . . .4986 .4990 .4857 .4890 . . . .4986 .4990 STA 2023 c B.Presnell & D.Wackerly - Lecture 12 is a standard normal r.v., then ¦ ¨ 0 P¥ ¨ ¡ Find 7 I5 $ 64 7 Ex. If 145 STA 2023 c B.Presnell & D.Wackerly - Lecture 11 146 Thought: If at first you don’t succeed, skydiving is not for you. 7 f5 $ 7 4 ¨ P¥  ¦ £ $f5 $ 7 4 ¨ ¨ 0 §¥ ¦ ¡ Find Assignments : Today : P. 215–226 For tomorrow: Exercises 5.24, 5.25, 5.31, 5.36–37,5.39–40, 5.95, £ 7 $f5 $ 64 £ ¨ ¦ P¥ 7 930 7 c P¥ 4 ¦ ¡ Find 5.96, 5.98 For Monday: Read pages 254–264, COMPUTER DEMO For Tuesday: Exercises 6.1, 6.3, 6.4, 6.8 £ £ 4 9"0 7 c P¥ ¦ 147 STA 2023 c B.Presnell & D.Wackerly - Lecture 12 STA 2023 c B.Presnell & D.Wackerly - Lecture 12 f & 64 ¨ Q (47   ¦P¥ ¡ 7 4 £ 94 BE 2301£BA¦ Last Time : Continuous Random Variables If is a STANDARD NORMAL random variable , find: ¡ Probabilities are areas under “density function” Find (p. 210). graph of f(x) P(a<X<b)  Q P¥ ¦ Find 0 f147 ¡  8¦§¨ ©¥£§¥ X8¤ a¥£P¥ £ ¨ ¦ £ ¦ ¨ Q ¦ ¦ Q 5§R ¨ £ P¥ 8§e Q £ P¥ ¦ £ ¦ Q ¦ Normal distribution is special case. pictures £ 0£ Key to finding correct areas (probabilities) : draw Q P¥ ¦ in Table IV, p. 809 .  ¡ and , for 0 ¡ Areas under normal curves between z-scores of £ 0 $f1 7 ¡ is a continuous r.v., (p. 210) £ 4¦ Q ( 7   §¥ If ¨ b £7 $  & 64 a 148 ¡ STA 2023 c B.Presnell & D.Wackerly - Lecture 12 have a standard normal dist.  §¨ Q  ¡ £ £  §¨ Q £ £  Q §¥ ¦ & 770 7 (5 $& 71d 70  Q P¥ £ ¦ such that . £ £ 70 X $& 71d ¦ P¥ ¨ ( 75 0 STA 2023 c B.Presnell & D.Wackerly - Lecture 12 152 Want - 98th percentile of dexterity scores. so that  £ ¨ 146@P¥ 5 74¦ £ $ 70 7 §¨ 70 £ ¡ ¨ 1 7 5 ¡ 4¦ @P¥ £BA £ ) 7 ( I3 RE 2 13 7£( . , its – Whatever the value of  – Find ¡ 151    P¥ ¦   §¥ ¦   f70 7 §¨ ¡ Find a number Employer requires dexterity score of at least 80.  f0 5 £ §¥ ¦ Want score on dexterity test Normal distibution, . Note : Used table in reverse. STA 2023 c B.Presnell & D.Wackerly - Lecture 12 Ex.: #5.31, p. 228 such that & (75 0 7  Q P¥ £ ¦ Find a number Recall the Empirical Rule!!! Find 150 Ex. Let Find 149 STA 2023 c B.Presnell & D.Wackerly - Lecture 12 score is score :  ¢¡ £ f0 5 £ P¥ ¡ ¦ £ score  £ £ £ $f0 5 £ §¥ ¦ (7 13 $ (0 0 £ ( 7( 13 $ ()4 ¡ ( 13 £ £ ¡ 153 STA 2023 c B.Presnell & D.Wackerly - Lecture 12 STA 2023 c B.Presnell & D.Wackerly - Lecture 12 Ex. Not in book ¡ If you randomly select 5 owners of garage door distance at which garage door opener is used. 7 0( 4 4 BE 2 0 %BA £ $£ £ ) Normally distributed, 154 openers and all 5 use the opener beyond 50’, what . do you think about the claim that 4 4 BE 2 0 %RA £ $£ What fraction use opener beyond the limit? 7  75 64 £ §¥ Y 7 ¦  Y 7 f6( £ §¥ Y 0 ¦ Ex. : Like #5.40, p. 229 (Same scenario, different numbers) Amount of dye, in ml., dispensed into can of paint is normally distributed, 7£ "& 7 RE something we can set £0 ¦ f6( £ §¥ What should we set unusable. ¡ STA 2023 c B.Presnell & D.Wackerly - Lecture 12 A , the score for 5 is .48 £ 0 z0 £( $ 7 70  £ P¥ ¦ must be approx. (£  2RA 57 )4 3& £ . ( score for ¦( I& 7 8 7 70  ¦ ¡ .5  Whatever the value of 155 milliliters Find to so that only 2% of cans are amount dispensed. A £¡ ¡ probability that all 5 used the openers beyond 50’? A ¡ £ BA More than 5 ml dispensed – paint unusable. Five users randomly selected. What is the  Thus, the ? – score : ¡ From table, Limit for interference: so that ¥ C7 70 2£ A & 7 ( £(  ¡ ...
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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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