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. Some courses 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,. 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 ).
Bostatstcs Math Revew 6) The arthm to ase of a postve umer s that umer whch satsfes the equato ad we wrte. We have 0 0 0.0 ; 0 (0.0) e e e e ; ( e ) e 0,04 ; (,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: 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: 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.) 0 Johs Hopks Uverst Departmet of Bostatstcs 08/8/
Bostatstcs Math Revew 0 Johs Hopks Uverst Departmet of Bostatstcs 08/8/
Bostatstcs Math Revew 0 Johs Hopks Uverst Departmet of Bostatstcs 08/8/ 4 0 0 5 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: D. More o Logarthms
Bostatstcs Math Revew 5 ) Wrte the followg equatos terms of arthms: 8 0 00 0 0.00 ) Wrte the followg equatos terms of epoets: 8 7 5 5 4 6 0 0.0 Commo arthm = a arthm wth ase 0 such that mples 0. Ofte 0 s wrtte as. Natural arthm = a arthm wth ase e such that mples e. Ofte e s wrtte as l. e 0 Note: e or Euler's costat (.788 ) s mportat descrg ocal relatoshps ad s useful ma statstcal applcatos. 0 Johs Hopks Uverst Departmet of Bostatstcs 08/8/
Bostatstcs Math Revew 6 Propertes of arthms: r r 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., 90.67.90670 0.000576.5760 4 ) 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:.4.79 4.9 8,000,000 0.4 0.079 0.00000049 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. 0 Johs Hopks Uverst Departmet of Bostatstcs 08/8/
Bostatstcs Math Revew 7 e.g., 9,800,000 7 9.8 0 has sgfcat fgures 0.0909 9.090 has sgfcat fgures 4 0.0005 5. 0 has sgfcat fgures a) Specf the umer of sgfcat fgures correspodg to each umer: 0.045 4.5 4.05 4.50 0.04 G. Roudg Correct to Decmal Places: the process of roudg a umer to decmal places. Rules for roudg (whe computg had) ) 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". (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: 7.865 7.847 7.85 7.875 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 = X + 0 Johs Hopks Uverst Departmet of Bostatstcs 08/8/
Bostatstcs Math Revew 8 where s the -tercept ad s the slope of the le ( the chage Y for each ut chage X). Suppose we have: Oservato X. 0.. 7 4. 4 9 5. 6 Y Plot each set of pots ad coect the pots a straght le. We would see the followg straght le (lear) relatoshp: 4 0 9 8 7 6 5 4 0 0 4 5 6 7 8 From the graph, we ca determe oth the -tercept ad the slope. The -tercept,, s the value of Y whe X=0. Here, =. The slope of the le,, 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 5 6 5 6 4 65 Page 9 4 5 5 6 9 4 5 9 0 Johs Hopks Uverst Departmet of Bostatstcs 08/8/
Bostatstcs Math Revew 9 8 7 0 9 8 5 45 7 8 5 7 0 5 5 5 844 6 9 4 6 4 6 56 5 8 0 Page 4 4 0 5 = 0 5 5 4 = 0 00 56 60 4 56 9 4 5 6 5 6 69 6 5 65 Note: Page 5 8 8 0 00 0 00 0 0.00 0 0.00 8 7 7 8 5 5 5 5 0 Johs Hopks Uverst Departmet of Bostatstcs 08/8/
Bostatstcs Math Revew 0 0 4 6 6 4 0.0 0 0. 0 Page 6.4.79 4.9 8,000,000 0.4 0.79 0.49.4 0 0.79 0 4.9 0 6 8 0.4 0.79 0 4.9 0 7 0.045 4.5 sgfcat fgures 4.5 4.05 4.50 4 0.04.0040 4 Page 7 7.865 7.86 7.847 7.85 7.85 7.85 7.875 7.88 0 Johs Hopks Uverst Departmet of Bostatstcs 08/8/