ࡱ> tvsijijNyy7I bjbj  hh5:#nn8M6-!######^p#}"}}#-@ }b8-! }! ~-d!BɉX V0!ΟΟΟݖ0vhT D}##}}}}Οn w: Section 7.1: Properties of the Normal Distribution Objectives: Students will be able to: Understand the uniform probability distribution Graph a normal curve State the properties of the normal curve Understand the role of area in the normal density function Understand the relationship between a normal random variable and a standard normal random variable Vocabulary: Continuous random variable has infinitely many values Uniform probability distribution probability distribution where the probability of occurrence is equally likely for any equal length intervals of the random variable X.. Normal curve bell shaped curve Normal distributed random variable has a PDF or relative frequency histogram shaped like a normal curve Standard normal normal PDF with mean of 0 and standard deviation of 1 (a z statistic!!) Key Concepts: Probability in a Continuous Probability Distributions: Let P(x) denote the probability that the random variable X equals x, then 1) The sum of all probabilities of all outcomes must equal 1 " P(x) = 1 ! the total area under the graph of the PDF must equal 1 2) The probability of x occurring in any interval, P(x), must between 0 and 1 0 d" P(x) d" 1 ! the height of the graph of the PDF must be greater than or equal to 0 for all possible values of the random variable 3) The area underneath probability density function over some interval represents the probability of observing a value of the random variable in that interval.  EMBED PowerPoint.Show.8  Probability of a Continuous Random Variable (from a Calculus Prospective): x=3 x=3 ( f(x) dx = 0.33 x ( = 1 x=0 x=0 Properties of the Normal Density Curve It is symmetric about its mean,  Because mean = median = mode, the highest point occurs at x =  It has inflection points at    and  +  Area under the curve = 1 Area under the curve to the right of  equals the area under the curve to the left of , which equals As x increases without bound (gets larger and larger), the graph approaches, but never reaches the horizontal axis (like approaching an asymptote). As x decreases without bound (gets larger and larger in the negative direction) the graph approaches, but never reaches, the horizontal axis. The Empirical Rule applies  EMBED PowerPoint.Show.8  Note: we are going to use tables (for Z statistics) or our calculator not the normal PDF!! Area under a Normal Curve The area under the normal curve for any interval of values of the random variable X represents either The proportion of the population with the characteristic described by the interval of values or The probability that a randomly selected individual from the population will have the characteristic described by the interval of values [the area under the curve is either a proportion or the probability] Standardizing a Normal Random Variable (our Z statistic from before) X -  Z = ----------- where  is the mean and  is the standard deviation of the random variable X  Z is normally distributed with mean of 0 and standard deviation of 1 TI-83 Normal Distribution functions: #1: normalpdf pdf = Probability Density Function This function returns the probability of a single value of the random variable x. Use this to graph a normal curve.Using this function returns the y-coordinates of the normal curve. Syntax: normalpdf (x, mean, standard deviation) #2: normalcdf cdf = Cumulative Distribution Function This function returns the cumulative probability from zero up to some input value of the random variable x. Technically, it returns the percentage of area under a continuous distribution curve from negative infinity to the x. You can, however, set the lower bound. Syntax: normalcdf (lower bound, upper bound, mean, standard deviation) #3: invNorm( inv = Inverse Normal Probability Distribution Function This function returns the x-value given the probability region to the left of the x-value. (0 < area < 1 must be true.) The inverse normal probability distribution function will find the precise value at a given percent based upon the mean and standard deviation. Syntax: invNorm (probability, mean, standard deviation) (the above take from  HYPERLINK "http://mathbits.com/MathBits/TISection/Statistics2/normaldistribution.htm" http://mathbits.com/MathBits/TISection/Statistics2/normaldistribution.htm) We can use -E99 for negative infinity and E99 for positive infinity. Example 1: A random number generator on calculators randomly generates a number between 0 and 1. The random variable X, the number generated, follows a uniform distribution Draw a graph of this distribution What is the P(0 0.95? Use calculator to generate 200 random numbers Example 2: A random variable x is normally distributed with =10 and =3. Compute Z for x1 = 8 and x2 = 12 If the area under the curve between x1 and x2 is 0.495, what is the area between z1 and z2? Summary: Normal probability distributions can be used to model data that have bell shaped distributions Normal probability distributions are specified by their means and standard deviations Areas under the curve of general normal probability distributions can be related to areas under the curve of the standard normal probability distribution Homework: pg 367 371; 7 12; 15-16, 19-20, 32-33 Section 7.2: The Standard Normal Distribution Objectives: Students will be able to: Find the area under the standard normal curve Find Z-scores for a given area Interpret the area under the standard normal curve as a probability Vocabulary: Z  the Z-score that corresponds to the area under the standard normal curve to the right of Z is  Key Concepts: Properties of the Standard Normal Curve It is symmetric about its mean,  = 0, and has a standard deviation of  = 1 Because mean = median = mode, the highest point occurs at  = 0 It has inflection points at    = -1 and  +  = 1 Area under the curve = 1 Area under the curve to the right of  = 0 equals the area under the curve to the left of , which equals As Z increases the graph approaches, but never reaches 0 (like approaching an asymptote). As Z decreases the graph approaches, but never reaches, 0. The Empirical Rule applies: 68% of the area under the standard normal curve is between -1 and 1. Approximately 95% of the area under the standard normal curve is between -2 and 2. Approximately 99.7% of the area under the standard normal curve is between -3 and 3.  EMBED PowerPoint.Slide.12  Example 1: Determine the area under the standard normal curve that lies to the left of Z = -3.49 Z = -1.99 Z = 0.92 Z = 2.90 Example 2: Determine the area under the standard normal curve that lies to the right of Z = -3.49 Z = -0.55 Z = 2.23 Z = 3.45 Example 3: Find the indicated probability of the standard normal random variable Z. P(-2.55 < Z < 2.55) P(-0.55 < Z < 0) P(-1.04 < Z < 2.76) Example 4: Find the Z-score such that the area under the standard normal curve to the left is 0.1. Find the Z-score such that the area under the standard normal curve to the right is 0.35. Summary: Calculations for the standard normal curve can be done using tables or using technology One can calculate the area under the standard normal curve, to the left of or to the right of each Z-score One can calculate the Z-score so that the area to the left of it or to the right of it is a certain value Areas and probabilities are two different representations of the same concept Homework: pg 381 383; 5 - 6, 9, 14, 21 22, 34, 37, 40 Section 7.3: Applications of the Normal Distribution Objectives: Students will be able to: Find and interpret the area under a normal curve Find the value of a normal random variable Vocabulary: None new Key Concepts: Finding the Area under any Normal Curve Draw a normal curve and shade the desired area Convert the values of X to Z-scores using Z = (X  ) /  Draw a standard normal curve and shade the area desired Find the area under the standard normal curve. This area is equal to the area under the normal curve drawn in Step 1 Procedure for Finding the Value of a Normal Random Variable Corresponding to a Specified Proportion, Probability or Percentile Draw a normal curve and shade the area corresponding to the proportion, probability or percentile Use Table IV to find the Z-score that corresponds to the shaded area Obtain the normal value from the fact that X =  + Z Example 1: For a general random variable X with  = 3 and  = 2 Calculate Z Calculate P(X < 6) Example 2: For a general random variable X with  = -2 and  = 4 Calculate Z Calculate P(X > -3) Example 3: For a general random variable X with  = 6 and  = 4 calculate P(4 < X < 11) Example 4: For a general random variable X with  = 3 and  = 2 find the value x such that P(X < x) = 0.3 Example 5: For a general random variable X with  =  2 and  = 4 find the value x such that P(X > x) = 0.2 Example 6: For random variable X with  = 6 and  = 4. Find the values that contain 90% of the data around  Homework: pg 390  392; 4, 6, 9, 11, 15, 19-20, 30 Section 7.4: Assessing Normality Objectives: Students will be able to: Draw normal probability plots to assess normality Vocabulary: Normal probability plot a graph the plots observed data versus normal scores Normal Scores is the expected Z-score of the data value if the distribution of the random variable is normal Key Concepts: Drawing a Normal Probability Plot (by hand) Arrange the data in ascending order Compute fi = (i 0.375) / (n + 0.25), where i 345B_ n o p ~ ` a b F G H W X Y Z [ Ḭypg_hp4CJaJh 5CJaJh 5CJaJhD^h 36CJaJh 3hc6CJaJhc6CJaJhqy 6CJaJh`P6CJaJh 3h 36CJaJhTlQ6CJaJh 3CJaJh CJaJhTlQCJaJh`PCJaJh`P5CJaJhD^h`PCJaJ hTlQ5 h`P5#45^ o p ~ b G H X Z m/$a$gde0gd q^`q q^`qgdTlQ[ v n $ B J Z lm+,-./0ǼǬǜǜǤǤqbZhp4CJaJjhqy hKCJUaJjQL hKCJUVaJhqy CJaJjhqy CJUaJhe0CJaJhTlQCJaJhICJaJh}CJaJh CJaJhd7hd7CJaJhd7CJaJh 5CJaJh}5CJaJhd75CJaJh{h 5CJaJhTlQ5CJaJ"/0~ p4k & Fgd9t$a$gd9t$a$$a$gdD) & F u^u`gdPZgdp4gd 01L\^ez}~PwiwaULhqy CJH*aJhe0he0CJH*aJhe0CJaJ j]hIhqy CJaJhIhqy CJaJ jhIhe0CJaJhIhe0CJaJhe0hqy CJH*aJhqy CJH*aJhqy hqy CJH*aJhe0he0CJH*aJh{5CJaJh{h{5CJaJhPZ5CJaJhd75CJaJhe05CJaJh{CJaJ "npŽsg^PD;h9t>*CJaJhXohXo>*CJaJh#hXo5>*CJaJh`P5CJaJh{hp45CJaJ jhD)h9t5CJUaJj%I hD)CJUVaJhD)5CJaJjhD)5CJUaJhACJaJh#jh#jCJaJh#jCJaJhXo5CJaJh#j5CJaJh{CJaJhe0CJaJhe0h{CJH*aJhe0hqy CJH*aJjvx`bp1żťŁyncUcGUh]h]5CJ \aJh2h25CJ\aJh25CJ\aJh]h]CJ aJhbCJaJh]h9tCJ aJh9tCJ\aJh9th9tCJ\aJmHsHh9th9tCJ\aJh9th9tCJaJh9t5CJaJh9tCJaJh9th9t5CJaJhKCJaJhXoh`P>*CJaJh9t>*CJaJh]>*CJaJp`bJKno)*op A B C  & Fgd#gd#gd2$a$gd2 & Fgd9t1T!IKYcd#$>?vw45=mo&')*oܹ}h6h6CJaJhVh60JCJ]aJh2h6CJ]aJh6CJ]aJjh6CJU]aJh2CJ]aJh2h2>*CJaJh2h25CJ\aJh26CJ]aJh2h2CJaJh2h26CJ]aJ0opz|  @ A D E \ ] _ ` z { } ~ """""""<">"R"T"^"ɾɾɾɾՉՉՀrrh#hc_CJH*\aJh#CJ\aJh#h#CJ\aJmHsHh#5CJaJh#h#5CJaJhKCJaJh#CJaJh#hc_CJaJh#hc_CJ\aJh#h#CJ\aJh#h#CJaJh#hK5CJaJh9tCJaJ)C D E ] ^ _ ` { | } ~ ""`"b"d"f"h"j" & Fgd#^gd# & Fgd#gd#^"`"h"j"""""####$#&#(#.#<#>#@###T$U$$$$$$&%'%)%1%2%ռՈՈՈ}tld\WR h<,5 h`P5h`PCJaJh^CJaJhICJaJh`P5CJaJhXohKCJaJh#hc_CJaJh#5CJ\aJh#hc_5CJ\aJh#CJaJhKCJaJh#h#CJaJh#hc_CJH*\aJh#hc_CJ\aJh#h#CJ\aJh#CJ\aJhc_CJ\aJ j"$#&#(#*#,#.#@##U$$$$'%X%Y%%%%,&.&J&''4''gdy( q^`q & Fgd#gd# & Fgd#2%7%J%W%X%Y%f%%%%,&.&J&L&N&' ''''4'6''++++++ƸƸƯ蝕shV#jL h~(h~(CJUVaJh~(h~(CJaJjh~(h~(CJUaJh8CJaJh#jhy(CJaJhy(CJaJhy(5CJaJh`P6CJaJhN 6CJaJhnVhnV6CJH*aJhnV6CJaJhvCJaJhxCJaJh`P5CJaJh`PCJaJ hv5 h=5 h`P5' (( )<) **++++++,5,6,7,A,B,C,L,M,N,W,X, & FgdgdgdA$ :;(a$gdY :;(gd% & F gdy(+++++++++*,+,4,5,6,7,@,A,B,C,K,L,M,N,V,W,X,Y,Z,f,,,,,,,,,,,,,¹wwwowgwwh2'CJaJh_CJaJhCJaJhhc_CJaJhhc_CJ\aJhhCJ\aJhhCJaJhh5CJaJh5CJaJh_h_5CJaJhIh`PCJaJhBCJaJjh~(h~(CJUaJj6h~(h~(CJUaJ(X,Y,Z,,,,,,,,,,,,,,,8-L-M-N-_-`-a-u-v- & Fgdgdqp & FgdgdgdA,,,,,,,7-8-K-L-M-N-^-_-`-a-t-u-w----;.=.D.E.F./// //////ĸĸ͡||sh`hICJaJhvuhvuCJaJh`P5CJaJhhc_6CJ]aJhhc_5CJaJh5CJ\aJhhc_5CJ\aJhCJ\aJhhCJ\aJh5CJaJh?U5CJaJhhc_CJaJhhc_CJ\aJhhCJaJhCJaJ$v-w-x----;.<.=.F.. /s/////5060_000000 q^`qgdqp & Fgd^gd & Fgdgdgdqp//0 0 0405060C0`00000000000000 1133344 4 4 4444547484[4\4h4i4444666P6廳ճh85CJaJh?U5CJaJh6hBU5CJaJh65CJaJh6h65CJaJh6CJaJhCJaJhqp6CJaJhU6CJaJhBUCJaJhy(CJaJhqp5CJaJ hy(5 hqp5hqpCJaJ.00001>1t2233i44526466666666gd8 & F"gd8 gd8 & Fgd6 & Fgd & Fgd6 ^`gd6gdqp q^`qgdqpP66666666666666666777777Z7\7h7j7~777ź󚒇~dQQ%h8h86CJ\]aJmHsH3h8h85B*CJ8OJPJQJ\^JaJ8phh85CJaJh8h`H]CJaJh?UCJaJh8h?UCJaJh8h`H]CJ\aJh8CJ\aJh8h,3CJaJh8h,3CJ\aJ%h8h,36CJ\]aJmHsHh8h86CJ\]aJh8h8CJ\aJ677777777777777788888888r9t9 gd8 & F#gd8gd8gdqp & F"gd8777777777777777777,8.8:8<8D8N8P8X8Z8p8r8|8~888888888999ʿ҅rri҅҅҅rrh8CJ\aJ%h8h,36CJ\]aJmHsHh8h86CJ\]aJh85CJaJh8h?UCJaJh?UCJaJh8h,36CJ\]aJh8h`H]CJaJh8CJaJh8h8CJ\aJh8h,3CJaJh8h,3CJ\aJh8h8CJaJ)999T9V9X9Z9`9b9p9t9999999999:<:>:@:B:H:J:X:Z:\:h::::::::B;D;F;H;J;P;R;ĹޝޝޑĹ~~~ޑĹvh?UCJaJ%h8h86CJ\]aJmHsHh8h85CJaJ%h8h,36CJ\]aJmHsHh85CJaJh8h?UCJaJh8h8CJaJh8h86CJ\]aJh8h8CJ\aJh8CJ\aJh8h,3CJ\aJ,t9v9x9z9|9~99Z:\:^:`:b:d:f:h:H;J;L;N;P;R;T;V;X;Z;;<gdBUgd8 gd8gdqpR;T;V;X;Z;p;v;;;;;<<<b<p</=>=`=j=k=======xxxxɱugu[uOMu[uUh6h66CJaJh6hO6CJaJh6hO6CJH*aJhOhOCJaJhSs5CJaJhO5CJaJhBU6CJaJhBU5CJaJhBUCJaJ hBU5h`PCJaJh CJaJhUCJaJhqpCJaJhqp5CJaJh8h&4CJaJh8h`H]CJaJh`H]CJaJh8CJaJ<<.<a<b<p<<.=/===>=k==xyeyfy#z$zUzVzmz & F)gdSsgdSs^gdO^gd6 & F*gdSs q^`qgdBU q^`qgdBUgdBU in the index (position of the data value in the ordered list) and n is the number of observations. The expected proportion of observations less than or equal to the ith data value is fi. Find the Z-score corresponding to fi from z-score table (Table IV) Plot the observed values on the horizontal axis and the corresponding z-scores on the vertical axis. If the sample data are taken from a population that is normally distributed, a normal probability plot of the observed values versus the expected Z-scores will be approximately linear. Drawing a Normal Probability Plot (Using TI-83) Enter raw data into L1 Press 2nd Y= to access STAT PLOTS Select 1: Plot1 Turn Plot1 ON by highlighting ON and pressing ENTER Highlight the last Type: graph (normality) and hit ENTER. Data list should be L1 and the data axis should be x-axis Press ZOOM and select 9: ZoomStat Does it look pretty linear? (hold a piece of paper up to it) Problem 10  EMBED PowerPoint.Slide.12   EMBED PowerPoint.Slide.12  Homework: pg 399 401; 3 8, 10 12 Section 7.5: The Normal Approximation to the Binomial Probability Distribution Objectives: Students will be able to: Approximate binomial probabilities using the normal distribution Vocabulary: Normal probability plot a graph the plots observed data versus normal scores Normal Scores is the expected Z-score of the data value if the distribution of the random variable is normal Correction for continuity 0.5, using a continuous density function to approximate a discrete probability Key Concepts: Criteria for a Binomial Probability Experiment: An experiment is said to be a binomial experiment provided: The experiment is performed a fixed number of times. Each repetition is called a trial. The trials are independent For each trial there are two mutually exclusive (disjoint) outcomes: success or failure The probability of success is the same for each trial of the experiment If X is a binomial random variable with np(1-p) e" 10, then we can use the area under the normal curve to approximate the probability of a binomial random variable. Use  = np and  = "np(1  p) as the parameters.  EMBED PowerPoint.Show.8  Remember we need to check using np(1-p) e" 10 before we can use these techniques.  Example 1: Consider a binomial variable X with n = 60 trials and p = 0.3 probability of success. This variable has Mean = n p = 18 and Standard deviation = " n p (1  p) = 3.55 Use a Normal approximation to estimate P(X d" 17) Example 2: Consider a binomial variable X with n = 100 trials and p = 0.05 probability of success. This variable has Mean = n p = 5 and Standard deviation = " n p (1 p) = 2.179 Use a Normal approximation to estimate P(X = 5) Summary: The binomial distribution is approximately bell shaped for large enough values of np(1 p) The normal distribution, with the same mean and standard deviation, can be used to approximate this binomial distribution (remember continuity correction value when converting) With technology, however, this approximation is not as needed as it used to be Homework: pg 406 407; 5 7, 15, 21, 28 Chapter 7: Review Objectives: xxxxdyeyfy"z#z$zVzlzmztzvzzzzzzzI{J{l{m{n{{{{{{{ŹxlX&j~L hSshSs5CJUVaJhSshSs5CJaJ jhSshSs5CJUaJhSshSsCJ\aJhSshSsCJaJhSsh,3CJH*\aJhSsh,3CJaJhSsh,3CJ\aJhSs5CJaJh?U5CJaJh 3 CJaJhOCJaJhOhOCJaJh6hO6CJH*aJmzzzzJ{m{n{{{{{{{{{{||||||||0||$a$gdSsgdBU`gdSsgdSs & F)gdSs{{{{{{{{{||||| |/|1|9|<|?||||||||||ɵ՛Ւzupupue\TLe\hvCJaJh`PCJaJh`P5CJaJh2h`PCJaJ hBU5 h`P5h6CJaJh 3 CJaJhBUCJaJhBU5CJaJh&45CJaJ jhSshSs5CJUaJ&j;L hSshSs5CJUVaJhSshSs5CJaJh?U5CJaJ jhSshSs5CJUaJ jqhSshSs5CJUaJ||||||K}}'~(~6~7~h~~~s$a$gdZI & F :;(gd :;(gd :(gd q^`qgd q^`q|J}K}}'~(~6~7~8~f~h~@ށûxjaRAja jhZIh?5CJUaJjFI hZICJUVaJhZI5CJaJjhZI5CJUaJh2hBU5CJaJh hBUCJaJh h CJaJh2h 5CJaJhVh 5CJaJh 5CJaJh CJaJh2h#5CJaJh`P5CJaJh2h`P6CJaJh:C6CJaJh`P6CJaJhBU6CJaJ‚$a$gd?:<Ă܂$&HJƒ(68Ƅ46D >?@ACDMTh&4hc_5CJ\aJh?CJ\aJh?h?CJaJh?h?6CJ\]aJh?h?CJ\aJh&45CJaJh?U5CJaJ%jh?5CJUaJmHnHuh?h?5CJaJh?5CJaJ5‚Ă8DEFGHIJKLMV & F gd&4gd&4gd?TUV+bc݇߇՜ٜ!$QSTݺñ{rp{h`h{r{r{{Xjh,3Uh$M'CJaJhCJaJUhv5CJaJhvCJaJ hK5 hv5h`PCJaJh CJaJhvuhvuCJaJh`P5CJaJh?U5CJaJh&4CJ\aJh&4hc_CJaJh&4hc_6CJ\]aJh&4hc_CJ\aJh&4hc_5CJaJh&45CJ\aJ!cއ3OhRSUVXY[\^_$a$gdv & F gd&4Students will be able to: Summarize the chapter Define the vocabulary used Complete all objectives Successfully answer any of the review exercises Use the technology to compute means and standard deviations of Continuous Probability Distributions Vocabulary: None new Homework: pg 408 411; 1, 4, 6, 8, 12, 17, 21, 23, 31, 32     Chapter 7: Normal Probability Distribution  EMBED PowerPoint.Slide.12  TVWYZ\]_hvCJaJj0h?UjGQL h2UVh2jh2UhI5CJaJhIjh,3Uh,3gdv,1h/ =!8"8#$% `!1}sz0nzV6h`ER41x} |TJ$Y@ @ n,$$H$!! 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'% %     V0KgvVKgvgvVKVKg&% % % " !% %   KcvZUkK%   6v%K6vK6vK6vK_6v_K-6v-K6vK6vK6vKg6vg% % " !% %   Uk&% ( Kg  6vg6vV6KV6Kg% % " !% %   &% ( Kg  6KVRpArial %w<' pjQ p4Q0PjQ luQDArialow' \PjQr$0' dv% GV6KVG%6K%G6KG6KG6KG_6K_G-6K-G6KG6KG6KGg6KgKV6vVKZ6KVZ6VZ6VYZ6YVZ6VZ6VhZ6hVZ6VZ6VvZ6vV% % % " !% % %   % % % " !% % %   Hd}\% % % " !% % %   HdzY% ( &% ( FMSGRBP  W4K6JKJWEc@o;w86FMSGREPFMSGRBPW@#6 641/,*(%#FMSGREPFMSGRBPW8##!FMSGREPFMSGRBPW0 FMSGREPFMSGRBPW<FMSGREPFMSGRBPWPk  "2CSZ`fkFMSGREPFMSGRBPW4kkqu}FMSGREPFMSGRBPW(FMSGREPFMSGRBP W(FMSGREPFMSGRBP W0FMSGREPFMSGRBP W,r|wrFMSGREPFMSGRBP W0_rrmhd_FMSGREPFMSGRBP W0K__ZUPKFMSGREPFMSGRBPW(7KKA7FMSGREPFMSGRBPW($77-$FMSGREPFMSGRBPW8$$"FMSGREPFMSGRBPW< FMSGREPFMSGRBPW@T !'/8@HOTFMSGREPFMSGRBPW4T`TWZ\^`FMSGREPFMSGRBPW4`q`cfjmqFMSGREPFMSGRBPW<qqv}FMSGREPFMSGRBPW0FMSGREPFMSGRBPW8FMSGREPFMSGRBPW4s }x sFMSGREPFMSGRBPKJ'%    V0HGNMKGNJKMHJKG% % % " !% % %   HdzYFMSGREPFMSGRBP6  V0|39369|63% % % " !% % %   HdzYFMSGREPFMSGRBP#  V0 & #&# % % % " !% % %   HdzYFMSGREPFMSGRBP  V0   % % % " !% % %   HdzYFMSGREPFMSGRBP  V0% % % " !% % %   HdzYFMSGREPFMSGRBP  V0% % % " !% % %   HdzYFMSGREPFMSGRBPk  V0hnknkhk% % % " !% % %   HdzYFMSGREPFMSGRBP  V0% % % " !% % %   HdzYFMSGREPFMSGRBP  V0% % % " !% % %   HdzYFMSGREPFMSGRBP   V0% % % " !% % %   HdzYFMSGREPFMSGRBP   V0% % % " !% % %   HdzYFMSGREPFMSGRBP r  V0ouoruro% % % " !% % %   HdzYFMSGREPFMSGRBP _  V0\b\_b_\% % % " !% % %   HdzYFMSGREPFMSGRBP K  V0HNHKNKH% % % " !% % %   HdzYFMSGREPFMSGRBP7  V04:47:74% % % " !% % %   HdzYFMSGREPFMSGRBP$  V0!'!$'$!% % % " !% % %   HdzYFMSGREPFMSGRBP  V0   % % % " !% % %   HdzYFMSGREPFMSGRBP  V0% % % " !% % %   HdzYFMSGREPFMSGRBPT  V0QWTWTQT% % % " !% % %   HdzYFMSGREPFMSGRBP`  V0]c`c`]`% % % " !% % %   HdzYFMSGREPFMSGRBPq  V0ntqtqnq% % % " !% % %   HdzYFMSGREPFMSGRBP  V0% % % " !% % %   HdzYFMSGREPFMSGRBP  V0% % % " !% % %   HdzYFMSGREPFMSGRBP  V0% % % " !% % %   HdzYFMSGREPFMSGRBP s  V0 pv ps v s p% % % " !% % %   HdzYFMSGREP% % % " !% % %   Hd}\% % % " !% % %   Rp ArialK<WQh00W  p݆04`Q. MJ\gK<5 Qow E\2r$0 Edv% % % % % % % % " !% % %   jD    Tf)AAL|Expected Z-score vs Time        Tl,>AA,LXValue  % % % % " !% % %   % % % " !% % %   % % % % " !% % %     TT9N@]AA9NLP0T`-@,AA-LT0.1T`-@AA-LT0.2T`-@AA-LT0.3T`-@AA-LT0.4T`-W@fAA-WLT0.5_T`-%@4AA-%LT0.6zT`-@AA-LT0.7T`-@AA-LT0.8T`-@AA-LT0.9TT9_@nAA9_LP1% % % " !% % %   % % % " !% % %     TTGbNqAAGbLP0YTTbqAAbLP5TXbqAAbLP10TXQb`qAAQbLP15TXbqAAbLP20TXbqAAbLP25TX`boqAA`bLP30TXbqAAbLP35TXb#qAAbLP40TXnb}qAAnbLP45% % % " !% % %   RpArial %w<' pkQ p4Q04kQ uQDArialow' \4kQr$0' dv% % % % % % " !% % %   v  TdzAAzLTTime % % % % " !% % %   % % % % % " !% % %   +  RpArial^w'f ud~wxwW|0* ,0*wX ,T\|$W|w%($w 3mw!%(Dwow u\W|r$0 udv% T$&AALlExpected Z-score     % ( % % % % " !% % %   % &% ( '% (     +Uk% % % " !% % %   Vj% % % " !% % %   Vj&% ( a   6aa'% (    V0^d^ada^   TYhAAY LdHours of TV  % % % " !% % %   Vj% % % " !% % %   % % % " !% % %   % ( % ( " !  " !  " @$@@4?D'D?D'D!b ! !!s % % %        : # ,,<@W@@$$,,<@W@@s Rp"System@5PI̕tV^S ;@%A ;@JĮw!"w5p@M&!ER%@@5pHPHPGPBPCP@* l*4|w#,wRw@P H 7C \* ȭ#P*GIwdv% F(GDICF MSGRCIQ    !" !" # ,,<@W@@$$,,<@W@@!  " !  " !  '% %     V0W- KgvgvVKVKg&% % % " !% %   KcvZUkK%   6v%K6vK6vK6vK_6v_K-6v-K6vK6vK6vKg6vg% % " !% %   Uk&% ( Kg  6vg6vV6KV6Kg% % " !% %   &% ( Kg  6KVRpArial %w<' pjQ p4Q0PjQ luQDArialow' \PjQr$0' dv% GV6KVG%6K%G6KG6KG6KG_6K_G-6K-G6KG6KG6KGg6KgKV6vVKZ6KVZ6VZ6VYZ6YVZ6VZ6VhZ6hVZ6VZ6VvZ6vV% % Rp"System@5PI̕tV^SArialow' \PjQr$0 dv*Ww̻*GIw̻*GIw`P@N%@N* >Iw!*=Iw!%!(*Iw!%dv% " !% % %   % % % " !% % %   Hd}\% % % " !% % %   HdzY% ( &% ( FMSGRBP  W4gxKJWEc@o;w86FMSGREPFMSGRBPW@r(~n 641/,*(%#FMSGREPFMSGRBPW8x.#!FMSGREPFMSGRBPW0 FMSGREPFMSGRBPW<aFMSGREPFMSGRBPWP.h  "2CSZ`fkFMSGREPFMSGRBPW4(~$kqu}FMSGREPFMSGRBPW(xFMSGREPFMSGRBP W(XFMSGREPFMSGRBP W0^FMSGREPFMSGRBP W,|wrFMSGREPFMSGRBP W0rmhd_FMSGREPFMSGRBP W0R7_ZUPKFMSGREPFMSGRBPW(1lXKA7FMSGREPFMSGRBPW(f7-$FMSGREPFMSGRBPW8+$"FMSGREPFMSGRBPW<%H7 FMSGREPFMSGRBPW@1 O !'/8@HOTFMSGREPFMSGRBPW4TWZ\^`FMSGREPFMSGRBPW41`cfjmqFMSGREPFMSGRBPW<+Bqv}FMSGREPFMSGRBPW0YIFMSGREPFMSGRBPW8T}FMSGREPFMSGRBPW4w|}x sFMSGREPFMSGRBPKJ'%    V0KGNJKMHJKG% % % " !% % %   HdzYFMSGREPFMSGRBP6  V0j]x369|63% % % " !% % %   HdzYFMSGREPFMSGRBP#  V0o8 #&# % % % " !% % %   HdzYFMSGREPFMSGRBP  V0  % % % " !% % %   HdzYFMSGREPFMSGRBP  V0% % % " !% % %   HdzYFMSGREPFMSGRBP  V0Wr% % % " !% % %   HdzYFMSGREPFMSGRBPk  V0 7/knkhk% % % " !% % %   HdzYFMSGREPFMSGRBP  V0o% % % " !% % %   HdzYFMSGREPFMSGRBP  V0% % % " !% % %   HdzYFMSGREPFMSGRBP   V0Nh% % % " !% % %   HdzYFMSGREPFMSGRBP   V0(% % % " !% % %   HdzYFMSGREPFMSGRBP r  V0oruro% % % " !% % %   HdzYFMSGREPFMSGRBP _  V0 "\_b_\% % % " !% % %   HdzYFMSGREPFMSGRBP K  V0(H@bHKNKH% % % " !% % %   HdzYFMSGREPFMSGRBP7  V0]u47:74% % % " !% % %   HdzYFMSGREPFMSGRBP$  V0!$'$!% % % " !% % %   HdzYFMSGREPFMSGRBP  V04  % % % " !% % %   HdzYFMSGREPFMSGRBP  V0(>?Y% % % " !% % %   HdzYFMSGREPFMSGRBPT  V0TWTQT% % % " !% % %   HdzYFMSGREPFMSGRBP`  V0`c`]`% % % " !% % %   HdzYFMSGREPFMSGRBPq  V0"|9qtqnq% % % " !% % %   HdzYFMSGREPFMSGRBP  V08S% % % " !% % %   HdzYFMSGREPFMSGRBP  V0Kb% % % " !% % %   HdzYFMSGREPFMSGRBP  V0n% % % " !% % %   HdzYFMSGREPFMSGRBP s  V0r  ps v s p% % % " !% % %   HdzYFMSGREP% % % " !% % %   Hd}\% % % " !% % %   Rp ArialK<WQh00W  p݆04`Q. 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F2 *Step 1: Verify normality assumption: is  -'',h"Arial?/?/?????? ???????/?/-.  2 np-'',h"Arial?/?/??????????????/?/-.  2 (1-'',h"Arial?/?/?????? ???????/?/-.  2 --'',h"Arial?/?/???????????????/?/-. 2 p) > 10?-'',h"Arial?/?/??????????????/?/-. 32 9500(0.75)(0.25) = 93.75 >> 10-'',h"Arial?/?/??????????????/?/-. 02 Step 2: Convert a to a Z-'',h"Arial?/?/??????????????/?/-.  2 --'',h"Arial?/?/??????????????/?/-. 2 score value, -'',hSymbol/?/?????????????/?/-.  2 3m-'',h"Arial?/?/??????????????/?/-.  2 <= -'',h"Arial?/?/?????????????/?/-.  2 Dnp-'',h"Arial?/?/???????????????/?/-. 2 S= 125 -'',hSymbol/?/??????????????/?/-.  2 os-'',h"Arial?/?/?????????????/?/-.  2 x= -'',hSymbol/?/??????????????/?/-.  2 -'',h"Arial?/?/??????????????/?/-.  2 np-'',h"Arial?/?/?????????????/?/-.  2 (1-'',h"Arial?/?/?????a????????/?/-.  2 --'',h"Arial?/?/??????????????/?/-. 2 p) = -'',hSymbol/?/??????????????/?/-.  2 -'',h"Arial?/?/?????? ????????/?/-. 2 93.75 = 9.6825-'',h"Arial?/?/??????????????/?/-.  2 Oa -'',h"Arial?/?/?????????????/?/-.  2 W--'',hSymbol/?/??????????????/?/-.  2 ]m-'',h"Arial?/?/???????????????/?/-. 2 a+0.5 -'',h"Arial?/?/??????????????/?/-.  2 -'',h"Arial?/?/???????????????/?/-. !2 125 80.5 -'',h"Arial?/?/??????????????/?/-.  2 -'',h"Arial?/?/??????????????/?/-. 2 125 -'',h"Arial?/?/??????????????/?/-.  2 --'',h"Arial?/?/???????????????/?/-.  2 44.5-'',h"Arial?/?/??????????????/?/-.  2 6Z = -'',h"Arial?/?/?????????????/?/-. 2 G ------------'',h"Arial?/?/??????????????/?/-.  2 t= -'',h"Arial?/?/??????????????/?/-. $2 }--------------------'',h"Arial?/?/??????????????/?/-.  2 = -'',h"Arial?/?/?????? ????????/?/-. 2 ----------------'',h"Arial?/?/??????????????/?/-.  2 = -'',h"Arial?/?/??????????????/?/-. 2 ---------'',h"Arial?/?/??????????????/?/-.  2 1= -'',h"Arial?/?/??????????????/?/-.  2 :--'',h"Arial?/?/???????????????/?/-. 2 =4.596--'',hSymbol/?/?????????????/?/-.  2 Ws-'',h"Arial?/?/??????????????/?/-. I2 ,9.6825 9.6825 9.6825-''',h,,h,,h,' @ )h'l 1#% EMFxy\KhCF, EMF+@XXFtEMF+@ @ 0@?!@ "@ *@$??@ ! 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