Week7 - STA 2023 c B.Presnell & D.Wackerly -...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: STA 2023 c B.Presnell & D.Wackerly - Lecture 11 Thought: “I personally think we developed language because of our deep need to complain” - Lily Tomlin Assignments : Today : pages 210-11, 215–226 For tomorrow: EXAMS RETURNED Exercises 5.15, 5.16, 5.19, 5.20, For Wednesday: Read more on pages 198–208 For Thursday: Exercises 5.24, 5.25, 5.31, 5.36–37, 5.39–40 133 STA 2023 c B.Presnell & D.Wackerly - Lecture 11 134 Last Time : The Binomial Probability Distribution   ¨  & 2¡ ¨ &¨ 652¡ 43 7 © ¤ 1 £ 2 3 ¢ § 98 ¢ ¢ © 1 & CB'''@(2 ¡!AAA!#! 8 for ¢ ..... 0 : (p. 1183) ¢ ¡ §© ¤ ¡ ¨ Tables : Contain © ¢ Variance (p. 185) : for ¢ £ Mean (p. 185) : § ¦¤ ¥ number of trials, ¡!&&&!#! 0)'('%$" If ¨ Binomial Experiment p. 179 ..... k k+1 n-1 n STA 2023 c B.Presnell & D.Wackerly - Lecture 11 135 Ex. Claimed Psychic. Five cards shuffled, one chosen. Psychic tries to identify card (without looking). Do this number of correct ¢ © 20 times and record identifications. Assume 20 trials are indep. correct ? ¢ § ¤ £ © # correct choices in 20 trys ¢ Let £ correct ¤ – § If the psychic is guessing, what is – What is a “success” in this case? ¥ ¢ is a binomial random variable with ¨ 2¡ ¢ 43 ¢ and ¨ and . ¢ © – . ¡ ¨ 2¡ 1 ¢ 1 ¢ ¢ 3 STA 2023 c B.Presnell & D.Wackerly - Lecture 11 136 What is prob. psychic is correct “6 or more § ¡ © ¤ £ times”? ..... 1 2 3 5 6 19 ¢ § ¢ ¡ © ¤ ¡ § 0 ..... © ¤ £ £ A A (Table II(h), p. 772) Discrete Random Variables Countable number of distinct values Look for “jumps” between the possible values. Assign a “chunk” of probability to each distinct value. 20 STA 2023 c B.Presnell & D.Wackerly - Lecture 11 137 Continuous Random Variables Take on values associated with points on one or more line intervals No gaps between values : Time, Height, Pressure, etc. Too many values go assign a “chunk” of probability to each Distribute our 1 unit of probability over appropriate interval like we would spread a handful of sand. – It will pile up in certain places. © – The depth or density depends on the value of . © – The “density” can be written a function of , (p. 210). §© ¤ A mathematical model for the population histogram STA 2023 c B.Presnell & D.Wackerly - Lecture 11 138 graph of f(x) P(a<X<b) b §© ¤ The total area under graph of is # a (p. 210). graph of f(x) Total Area = 1 Over a given interval, the area under the graph of § ¤ © a value in that interval (p. 210). – How do I find these areas? Calculus (not in this class) Use a table. © corresponds to the probability that will take STA 2023 c B.Presnell & D.Wackerly - Lecture 11 for all , § ¢ © ¤ ¢ £ Note: 139 Line has area 0 so, for contin. r.v.’s, “endpoints” don’t make any difference in probs. of intervals, unlike discrete a (p. 210). is a continuous r.v., § ' 7 ¡ § ( ¡ ¢ © ¢ ¤ £ © ¢ § ( ¤ ¢ ¡ £ ¢ § ( 7 ¡ © ¢ 7 ¤ £ © ¢ 7 ¤ £ © If STA 2023 c B.Presnell & D.Wackerly - Lecture 11 140 The Normal Distribution A random variable has a normal dist. with mean if its probability density is (p. 216) ¨ ©4 § ¥ ¦§   ¤ ¤ £ ¢ ¡ # 3 ¢ 3 and std. dev. 1 One very important continuous distribution. §© ¤ Recall:   # & ¢ # & 9# " && '(& £ ¢ ¡ Formula traces out the “bell shaped curve”. Will write && ''& No need to know this formula. © As a shorthand for “ has a ! © and std dev ” MEANS “ has a normal distribution #& and std dev § #&  !  ¤ ! © with mean © N 1 3! 1 ¤ normal distribution with mean 3 § N STA 2023 c B.Presnell & D.Wackerly - Lecture 11 141 The graph of the normal density is a symmetric, . The std dev, 3 1 bell-shaped curve, centered at , measures the spread of the distribution. (See Figure 5.6, p. 216) µ =8 σ =1 4 µ =12 σ =1 6 8 10 12 14 16 Two Normal Curves with Diff Means, Same Std Dev µ =10 σ =1 0 µ =10 σ =3 5 10 15 Two Normal Curves with Same Mean and Differnent Std Devs 20 STA 2023 c B.Presnell & D.Wackerly - Lecture 11 142 The Standard Normal Distribution – p. 217 is said to have a standard normal and std dev , & © ¢ 3 #! $" ! ¡ The -score gives the number of standard . © is larger than © is smaller than . . 1 1 is from 1 © deviations . 1 ¢ ¢ £ ¢ £ © ¢ £ is a standard normal random variable: ¢ . is already a -score. © # © ¢ 1 ¤ 3 © ¢ That is, the value of © ¤ © If 1 & ¤ Recall : -score : ¤ § i.e., is normal with mean distribution if # The r. v. STA 2023 c B.Presnell & D.Wackerly - Lecture 11 143 For any normal random variable, areas under normal are ¢ and , for curves between z-scores of given in Table IV, p. 809 . Tabled area 0 z Given in table 1/2 1/2 -z 0 Find 0 Find z Find STA 2023 c B.Presnell & D.Wackerly - Lecture 11 144 (Part of the) Standard Normal Table 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 11 is a standard normal r.v., then ¢ & § ¤ £ ¢   @# & ¢ ¤ ¢   %# & & § ¢ £   @# & § ¢ ¤   @# & £ ¤ ¢ Find Find ¢ Find § Ex. If 145 £ ¢ & § ¢ ¢ § # 2 & # & ¤ ¤ £ £ STA 2023 c B.Presnell & D.Wackerly - Lecture 12 146 Thought: If at first you don’t succeed, skydiving is not for you. Assignments : Today : P. 215–226 For tomorrow: Exercises 5.24, 5.25, 5.31, 5.36–37,5.39–40, 5.95, 5.96, 5.98 For Monday: Read pages 254–264, COMPUTER DEMO For Tuesday: Exercises 6.1, 6.3, 6.4, 6.8 STA 2023 c B.Presnell & D.Wackerly - Lecture 12 147 Last Time : Continuous Random Variables Probabilities are areas under “density function” (p. 210). graph of f(x) P(a<X<b) a b If Normal distribution is special case. Areas under normal curves between z-scores of is a continuous r.v., (p. 210) § ( 7 ¡ § ( ¡ ¢ © ¢ ¤ £ © ¢ § ' ¤ ¢ ¡ £ ¢ § ( 7 ¡ © ¢ 7 ¤ © £ 7 ¤ £ ¢ ¢ and , for in Table IV, p. 809 . Key to finding correct areas (probabilities) : draw pictures STA 2023 c B.Presnell & D.Wackerly - Lecture 12 is a STANDARD NORMAL random variable , find: § ¢ § ¢ 7  @# & ¢ 7  @# & 32 ! & ¤ # ¢ ¤ £ # & ¤ ¢ ¤ 1 Find # Find § If 148 £ ¢ § ¢ ¢ §  &  ¤ 7 & ¤ 7 ¤ £ ¤ £ ¤ STA 2023 c B.Presnell & D.Wackerly - Lecture 12 § § ¢ ¢ ¢ ¤ 7 ¤ 7 ¤ ¤ £ Find 149 £ ¢ Recall the Empirical Rule!!! § & § ¢  & ¢ ¢ ¢  & ¢ & @# ¤ ¢  £ & %# Find ¤ £ STA 2023 c B.Presnell & D.Wackerly - Lecture 12 have a standard normal dist.    & 7 ¢ ¤ £ such that .    & § ¢ 7 Find a number § Ex. Let 150 ¤ £ ¢ Note : Used table in reverse.  & ¢ § 7 ¤ £ such that ¡ § ¡  & ¢ ¢ Find a number . ¢ ¤ £ STA 2023 c B.Presnell & D.Wackerly - Lecture 12   & ¢ 3! ¢  ¢ 1 Normal distibution, score on dexterity test & © Ex.: #5.31, p. 228 151 Employer requires dexterity score of at least 80. © ¤ ¢ @ £ ¤ score : § Want . ¢ ¢ ¢ § § © ¤ ¢   ¢ © ¤ £ £ STA 2023 c B.Presnell & D.Wackerly - Lecture 12 152 Want - 98th percentile of dexterity scores. so that , its © score is ¢ ¢ score ¢ ¢   2  & ¢    & #  © – Whatever the value of ¤ – Find   ¢ © ¤ STA 2023 c B.Presnell & D.Wackerly - Lecture 12 153 Ex. Not in book distance at which garage door opener is used. ¢ ## ¢ 3!  . & ¢ 1 Limit for interference: © Normally distributed,  What fraction use opener beyond the limit? § & © ¤ ¢ % £ & § ¢ § ¢  %# &  ¢ ¤ score : © ¤ ¤ £ £ Five users randomly selected. What is the probability that all 5 used the openers beyond 50’? STA 2023 c B.Presnell & D.Wackerly - Lecture 12 154 If you randomly select 5 owners of garage door openers and all 5 use the opener beyond 50’, what do you think about the claim that ? ## ¢  3! ¢ 1 – Ex. : Like #5.40, p. 229 (Same scenario, different numbers) Amount of dye, in ml., dispensed into can of paint is normally distributed, & & ¢ 3 something we can set ¢ 1 More than 5 ml dispensed – paint unusable. 1 What should we set to so that only 2% of cans are unusable. © Find amount dispensed. so that 1 ¢ STA 2023 c B.Presnell & D.Wackerly - Lecture 12 , the ¤ 1 Whatever the value of 155 score for 5 is ¢ .5 .48 0 & ¤§  & ¤  ¤ #&  1 ¢ ¢ & ¤ £  ¤ £ . must be approx. §  ¢ score for ¢ Thus, the § ¢  & From table, z0 milliliters ¢ ¤ 1 & ...
View Full Document

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.

Ask a homework question - tutors are online