Johns Hopkins University Department of Biostatistics Math Review for Introductory Courses
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1 Johs Hopks Uverst Departmet of Bostatstcs Math Revew for Itroductor Courses Ratoale Bostatstcs courses wll rel o some fudametal mathematcal relatoshps, fuctos ad otato. The purpose of ths Math Revew s to provde a opportut for ou to revew some asc deftos, relatoshps ad perform some had calculatos. Bostatstcs 60 wll requre use of a had-held calculator for homework ad eamatos. I the Bostatstcs 6-64 seres, the statstcal aalss package Stata wll also e used. A. Deftos Some asc deftos ad omeclature are useful. ) Itegers are whole umers, oth egatve ad postve: (, -, -, -, 0.,, ) ) Postve tegers are tegers greater tha 0: (,,, 4, ) ) Real umers are all umers, oth tegers ad otegers such as fractos. 4) a s called the asolute value of the umer a: If a 0, the a a If a 0, the a a For eample, ad. 5) If s a real umer ad s a postve teger, the tmes. I other words, K. s defed as multpled tself multpled tself tmes Here s referred to as the ase ad as the epoet. We ca wrte 0 ( coveto) f s a postve teger (the th root of ).
2 Bostatstcs Math Revew 6) The logarthm to ase of a postve umer s that umer whch satsfes the equato ad we wrte log. We have ; log 0 (0.0) e e e e ; log ( e ) e 0,04 ; log (,04) 0 7) The summato sg s a useful otato to dcate the sum of the values of a varale for oservatos through. Let represet some varale such as age. We ca let dcate the value of age for dvdual, where takes o values from to, a group of dvduals. The sum of the ages of all dvduals ca e wrtte as: + + K+ 8) The product sg s a useful otato to dcate the product of the values of a varale for oservatos through. Usg the prevous eamples, the product of the ages of all dvduals ca e wrtte as: K B. Eamples of Usg Summato Sgs Suppose we have oservatos, each descred a varale ad a varale. Let 5 9 ad 6 4 For the values of ad gve aove, compute the followg pecl ad paper just for the metal eercse, ot calculator. ( You ca later verf our results usg our calculator.) 000 Johs Hopks Uverst Departmet of Bostatstcs 06/08/0
3 Bostatstcs Math Revew 000 Johs Hopks Uverst Departmet of Bostatstcs 06/08/0 ( ) + ( ) + ( ) ( ) ( )
4 Bostatstcs Math Revew 000 Johs Hopks Uverst Departmet of Bostatstcs 06/08/0 4 0 ( ) 0 5 log log C. More Eamples Compute the followg commol used equatos had: The followg otato s used for the sample mea: The two equatos elow are algeracall equvalet formulatos of the sample varace: ( )
5 Bostatstcs Math Revew 5 D. More o Logarthms ) Wrte the followg equatos terms of logarthms: ) Wrte the followg equatos terms of epoets: log 8 7 log 5 5 log ( ) 4 6 log 0 ( 0.0) Commo logarthm a logarthm wth ase 0 such that log 0 mples 0. Ofte log 0 s wrtte as log. 000 Johs Hopks Uverst Departmet of Bostatstcs 06/08/0
6 Bostatstcs Math Revew 6 Natural logarthm a logarthm wth ase e such that log e mples e. Ofte log e s wrtte as l. Note: Propertes of logarthms: e or Euler's costat (.788 ) s mportat descrg ologcal relatoshps ad s useful ma statstcal applcatos. ( ) log log ( ) log log log + log log r ( ) r log E. Scetfc Notato: Epressg a umer as a product of a umer N etwee ad 0 ad a tegral power of te order to smplf otato of calculatos: ( N ) ( 0) k (e.g., ) Rules ) The epoet of 0 s determed coutg the umer of places that the decmal pot was moved whe gog from the orgal umer to the umer etwee ad 0. ) The epoet s a) egatve f the orgal umer s less tha ) postve f the orgal umer s greater tha 0 c) 0 f the orgal umer s etwee ad 0 Epress the followg scetfc otato: ,000, Johs Hopks Uverst Departmet of Bostatstcs 06/08/0
7 Bostatstcs Math Revew F. Sgfcat Fgures: A dgts a umer whch cotrute to the specfcato of ts magtude apart from zeroes that determe the posto of the decmal pot. HINT: It helps to frst wrte the umer scetfc otato order to determe the umer of sgfcat fgures. e.g., 9,800, has sgfcat fgures has sgfcat fgures has sgfcat fgures a) Specf the umer of sgfcat fgures correspodg to each umer: G. Roudg Correct to Decmal Places: the process of roudg a umer to decmal places. Rules for roudg ) Roud to the earest umer ) If the umer determg the roudg s 5, set a polc to alwas roud to the eve umer (or alwas to the odd umer) to mmze overestmato or uderestmato. Ths s mportat for had calculatos. If ou alwas roud "up", our calculatos ma ted to e overestmates. B choosg to roud to the eve umer, we would roud.45 to.4,.5 to.4, 4.5 to 4., 5.75 to 5.8. The rouded values are correct to decmal place. I ths wa, / of the tme we roud "up" ad / of the tme we roud "dow". 000 Johs Hopks Uverst Departmet of Bostatstcs 06/08/0
8 Bostatstcs Math Revew 8 (eg,.459 s.4 correct to decmal places; 7.45 s 7.4 correct to decmal place) a) Correct the followg umers to two decmal places: H. Equato for a Straght Le: Suppose ou ve collected depedet pars of data (X, Y ), to, for oservatos. Suppose we let Y represet our outcome of terest, ad X some fed cotuous covarate. Is there a perfect lear relatoshp etwee Y ad X; that s, do the pots fall eactl o a straght le? If so, we could wrte: Y βx + α where α s the -tercept ad β s the slope of the le ( the chage X for each ut chage Y). Suppose we have: Oservato X Y Plot each set of pots ad coect the pots a straght le. We would see the followg straght le (lear) relatoshp: Johs Hopks Uverst Departmet of Bostatstcs 06/08/0
9 Bostatstcs Math Revew 9 From the graph, we ca determe oth the -tercept ad the slope. The -tercept, α, s the value of Y whe X0. Here, α. The slope of the le, β, s the chage the value of Y for each oe ut chage X. The slope ca e derved from a two pots o the straght le ad s equal to )Y/)X or (Y-Y)/(X- X). Usg the pots for oservatos ad, we ca calculate the slope as (-)/(-0). The, we ca epress the lear relatoshp etwee X ad Y the equato of the straght le Y X +. Math Revew for Bostatstcs Courses Aswer Ke Page Page ( ) 8 ( ) ( ) ( ) ( ) ( ) 9 ( ) 4 + ( 6) + ( ) ( ) ( 4) + ( 6) + ( ) Page Johs Hopks Uverst Departmet of Bostatstcs 06/08/0
10 Bostatstcs Math Revew 0 log ( ) log 0 ( 5 5 4) log 0 ( 00) log log ( ) ( ) ( ) Note: Johs Hopks Uverst Departmet of Bostatstcs 06/08/0
11 Bostatstcs Math Revew Page 5 8 log log log log log log log ( 6) 4 ( 0.0) Page ,000, sgfcat fgures Page Johs Hopks Uverst Departmet of Bostatstcs 06/08/0
12 Bostatstcs Math Revew 000 Johs Hopks Uverst Departmet of Bostatstcs 06/08/0
Johns Hopkins University Department of Biostatistics Math Review for Introductory Courses
Johs Hopks Uverst Departmet of Bostatstcs Math Revew for Itroductor Courses Ratoale Bostatstcs courses wll rel o some fudametal mathematcal relatoshps, fuctos ad otato. The purpose of ths Math Revew s
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