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# STATISTICS MATH 203

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This 19 page Class Notes was uploaded by Sadie Schroeder on Wednesday September 30, 2015. The Class Notes belongs to MATH 203 at Western Kentucky University taught by David Neal in Fall. Since its upload, it has received 18 views. For similar materials see /class/216741/math-203-western-kentucky-university in Applied Mathematics at Western Kentucky University.

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Date Created: 09/30/15

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are WALhm utia wn 55 in us an IQ some Ewe ubtmrl n rmnn evzxy measurnnen39 men we ullhave anmmal dmnhuuan hutmnxt values wd he hetwem ex and 2 By mmdenwunenendm m a an euppdee we dende me new values by 41 en between 1 and 1 Tnenenm 1s ndw Nm 1 By sumamng n rmnn every measureman and men dwmmg by a we have eundndmd the values and have ahtamed une standard mxmd dmnhuudn 2 Nm 1 LetXNUhaJandZ Then 2 N W 1 Fax 2 Nm 1 we sun can ennpme une vanaus pxahahlhues and Inverse calmlaumsunrgthe mmnlod and innNam mmmands Fm example PZZ 4 22 and mazsszsms reshvwnhelww 233 39 39 SETSE 39 39 EEE767699 sssevdezvs e 1 zNmn H2422 PHizsszsmS Bangle Let 2 Nm 1 a FmdLhenumbex z urhthal Fuzz nus h FmdLhenumhzrsw and z menunamwszsz n95 snddunn Fax Part a we actua y need PZ z n 95 Sa enter me mmmand invNamI as n n m ablam 1m n mnx n ressssssds nvNurMS75v vl x95995 Fax Part h we need w and 1 men that PZ w ms and w stz n 915 5d enter me mmmands mmmnmzs n n and mmewmnzsmmdmnw nemnnd isms D Neal 5pm 2cm By mnv mm Mum narma dmnhuuans X and Y in man narma mmmuuans um X and Y canhe placed an m m scale Values mm X and Y um canhe mnparedwnhwutany pxahahxhty madman Bangle IQmm are X Mlm15 and baby mm mm are Y 2 lbs thrhxslesslmdy an Q Hi at Ma Ms Dr anewrhmn wexghmg at Ma m paunds7 mum Simply canven ch value in a mammxm ml 45 gtZZZ Yzmazz gtZZIS szazz 15 Thexange 223 mm m pxahahdny than 221mm an Q a at Ma Ma Isles likely than anewham wnghmg at 1m 1m paunds Dr Neal Spring 2009 Practice Exercises 1 Students in a Psychology Masters Program are given an IQ test The scores are generally found to be normally distributed with a mean of 112 and a standard deviation of a Give the population Q under consideration and the measurement X What is the notation for the distribution of X 7 b Compute i PX s 105 3911 P100 3 XS 124 iii PX z 130 c What scores x and y symmetric about the mean are such that Px SX S y 066 7 2 Let z N N01 a Compute 391 PZZ 145 3911 PZs 212 iii P 2 3 ZS 2 b Find the numbers w and 2 such that Pw S Z S2 097 c Find the number 2 such that PZ 2 z 001 3 ACT scores are X N224 32 and SAT scores are Y N1020 160 a Which is a better score an ACT of 28 or an SAT of 14007 b Which happens less often an ACT of at most 14 or an SAT of at least 14007 4 A sample of 100 watt GE light bulbs are tested for lifetime during everyday use The lifetimes in hours before burning out were as follows a Give the population Q under consideration and the measurement X b Find the sample mean and deviation c What percentage of these measurements are within 7c i S 7 Within c i 2S7 Do these times appear to be normally distributed7 D Neal 5pm 2cm 1 r2 Al 1udmtsmthsl7xythnlugy Mmquot ngram X Q 82912 N mm 9 h HXsms mas Pmn X 124znxns77 KXZIZEIMHHZNS a x mMarm1nzazxnsmsanay invNan xSAlLEleznsms 2 a H22145znn1353 stizlzp n11 mzszsmnms invmmms n n and invNamIWX n n 39vaman 1122326 Hmwzwazzm Sn an SAT a Y Mon pmduczsdqehghzx scare an a mmquot scale quotgnaw mmwwamm Wm ZS m mmpmmmm 11 MM 222375 SH mmmammmmwmm h XSMAZS A a n A l wattGEhghththsdunng39hspxaduman X 7 human th h sxxsaxmszmns a may sshmam xsshrs whmh mntamsA AS m mm x199 uqueumes 7 2 was 252 mm m an hrs whmh mnmnsASAS m mm 37 x99 fth umes Ahmagram shaws39hatdqe me are nut at a nmma y dmnhuted Dr Neal Spring 2009 MATH 203 Basic Statistics Let Q be a population under consideration and let X be a specific measurement that we are analyzing For example Q All US households and X Number of children under age 18 living in the household To study this scenario we obtain a set of measurements x1 x2 xn which may be either a census or simply a random sample Census In a census we assume that we have a measurement from every member in the population under consideration For small populations such as students in one particular class or players on one sports team it is not hard to obtain a census by surveying each person in that population But for extremely large populations such as all US households it is nearly impossible to obtain a real census even when mandated to do so every ten years by the United States Constitution But when we do have a census of measurements X from a population then we can find the true values of the mean u the variance 02 the standard deviation 0 as well as other population parameters Mean Given a set of n measurements xlx2xn the mean or average of these specific values is given by 7 x1x2xn n When the values are a census of a specific measurement X from a population Q then p is true average value It is also called the expected value of X and may be denoted by X or EX Variance The variance denoted by 02 is the average squared distance from the mean and is given by 2 2 2 n 2 x x x 1 2 o 1 H 2 3 n Zxilquot il Alternately the variance is the average of the squares minus the square of average and can be computed by 2 x12 x22xn2 2 7 p n The variance is sometimes denoted by a or VarX D Neal 5pm 2cm We lathe square mat anhe variance in genhe standald deviadnn denated by a The standard mam Qves away armeasunhg nae averagesprmd nm he man A we a means e a Lhatmeaxur mems are mmls ndy dnseln zheamge a Median mdMode When he measuremehas hm m are m meashg Urdu than he aedsah Ts nae mddh value ax nae Vang anhe twammdhvaluesn than are ah even number a sure has The made Ts ua mea me e measurement at measurermnm that Damn mast eh Bangle x Belaw are nae number a rmm hams enm ed m Lhm semesm fax a student m ahe seem a MATH 115 Fmd nae mean variance standard dwnlmn meshah and made athesevaJues Nhatpzmmtage armesemeasmemems are WALhm ahe standard dewauan a class averag cmhamamm same snjhunh Let n thss eu cMATII 115 class and let X Numhzr a rmm hams mm ed m Lhm semesm Bemuse we have a eehsus m Lhm class we can nd nae hue mean a ahduaehue standard dmauana Ta an sq we shan enter nae datamta nae calculatax smutmtamueanrg mdex and use nae LVus Stats mmmand Enterdztzinw u Sandal Lhanenker LVzr Stats u 121512 15 576 Themzanxsu e376 T ememnhn Note The calmhtax aespxays39hesvasue as 7 wheh standsfm mplemmn Buthecause wehave a eehshsamaesdassaha natmznly a sample we use a m Iszesentdqat we have nae real average m a 15 rudnhaurs Dr Neal Spring 2009 The variance is computed by 2 2 2 9324 02 x1 x2 xquot 2 1623 n 36 Then taking the square root gives us the standard deviation of o J3 m 1732 The true standard deviation is displayed as ox on the calculator output and this value is to be used if we have a census of measurements So now we can say that the class average is 16 credit hours with an average spread from 16 of 0 m 1732 credit hours The median is the middle measurement But because we have an even number of measurements 36 we must take the average of the middle two measurements After sorting the 36 values the middle values are in the 18th and 19th positions The 18th value is 15 while the 19th value is 16 So the median is 15 16Z 155 which is also displayed on the TI After sorting the values it is easy to make a frequency chart from which we see that the mode is 18 hours That is in this class more students are registered for 18 hours than for any other number of hours Hours Students To find the pct within one standard deviation of average we first compute u i o 16 i 1732 which is about 14268 to 17732 So students taking 15 16 or 17 hours fall in this range There are 10 3 3 16 students in this range Thus 1636 or 4444 of the students in this class are within one standard deviation of class average Question Is this class representative of all students on campus Representative of just undergraduates Representative of all students taking a Gen Ed math class this semester Or perhaps representative of just MATH 116 students this semester Probably the most we can say is that this class is representative of all MATH 116 students this semester If you want a sample that is representative of a larger portion of the student body then you must sample accordingly from among that entire group of students But you should never take an existing sample and try to say that it is representative of a larger group that was not represented in the sample Dr Neal Spring 2009 Sample Mean and Sample Deviation Often a collection of measurements is just a sample from a larger population In this case we cannot find the real average It Instead we can only compute the sample mean denoted by TC However 7c is computed the same way as we computed p by adding up the values and dividing by n we just denote it now by TC to specify that we are only working with a sample The sample deviation denoted by S is computed similarly to 0 however we use 9 in the formula rather than u and we average the squared differences by dividing by n 1 rather than n 1 n 2 2 Ln Z 0 quot 06 M S n1 x1 x r 1 F0 a census For a sample 1 By dividing by n l the sample variance S2 becomes an unbiased estimator of the true unknown variance 02 That is the average of all possible S2 from all possible samples of size n will equal the true variance 02 Quartiles and 15 IQR The first quartile Q1 is the median of just the measurements that are below the overall median The third quartile Q3 is the median of just the measurements that are above the overall median These values are displayed along with the minimum median and maximum in the 17Vars Stats output Together the values min 7 Q1 7 med 7 Q3 7 max make up the five7number summary The 15 IQR or 15 Interquartile Range is the interval Q1 15 X Q3 Q to Q3 15 gtlt Q3 Q1 Values from a sample that are outside this range are called outliers and are often excluded from samples so as not to throw off the average too much Example 2 Below are data on city mpg from a sample of two7seater cars Model City MPG Model City Acura NSX 17 Honda Insight 57 Audi TT Quattro 20 Honda 2000 20 Audi TT Roadster 22 Lamborghini Murcielago 9 BMW M Coupe 17 Mazda Miata 22 BMW Z3 Coupe 19 Mercedes7Benz SL500 16 BMW Z3 Roadster 20 Mercedes7Benz SL600 13 BMW Z8 13 Mercedes7Benz SLKZ 30 23 Chevrolet Corvette 18 Mercedes7Benz SLK320 20 Chrysler Prowler 18 Porsche 911 GT2 15 Ferrari 360 Modena 11 Porsche Boxter 19 Ford Thundgrbird 17 Toyota 2 25 D ea 5pm m Us yaur calmlatm faith rnurwrrg a Fmd the sample mean and sample dewauan the rneaun the made and the wee number Summary What percentage nnnese mueages are WALhm rne sample dwnuan m ungxe weuge2 11 Make ahntugramwnhnrge m 5 an dxwdedmta hm aflenthS Wuhhmhas the rnnnrnemrernenm Tne semnd max m cweune I 5 QR and denme any suxpnled rumeu SnluLLnn x We rm mm the lama 1th the STAT EDIT Susan Fax Lhm pmhlemwe shalluse L2 Aner entnxrgthe am we smith am wALh the mmmand SonA J Then we enrnpme the shame wALh the mmmand LVu Stats L2 Enter am inw L2 Sound mwubstzs Because me am are Duly a mp rTrnernerernenn mm une pvpulauun n n all rwreeener me n cars une value n2 7 e 1555 me mplemean The sample dmaumxsmxplayedasl39z ll ernnernurn uuss vmtahe whdedqe mammumvalue 1s 51 Themednn 1s gwen 135 Thain IX 51 me average rmne Wu mddh maimemznts when Ln nrrernrg axdzr the men and 12m wALthm uu set n2 eyennze 22 The men value I lxwhle39helquvaluusl 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suntan a mzaxuremznts There are nu anthers In 2 AHWKUFnshmzn Dr Neal Spring 2009 MATH 203 Confidence Interval for Difference of Means of Normally Distributed Measurements Here we discuss an improvement for a confidence interval for the difference of means of two normally distributed measurements both of which have unknown standard deviations The results are based on independent random samples of sizes ml and n2 respectively taken on each measurement For calculations on the T1783 84 we can use the ZiSampTInt screen from the STAT TESTS menu This feature require that we specify whether or not we wish to use the pooled deviation Sp We should specify Yes only under the assumption that the two measurements have the same standard deviation When the normal measurements are assumed to have the same unknown deviation then the pooled variance and standard deviation estimates are n1 1S12 n2 1S and n1 1S12 n2 1S22 S2 p n1n2 2 n1n2 2 The confidence interval for the difference in means is now given 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