LINEAR ESTIMATES
1. Find Find an approp appropria riate te discret discretee data to be analyzed. analyzed. The The data shoul shouldd contain contain more more than 10 pairs pairs of values. A study was conducted to determine the effects of sleep deprivation on student's ability to solve problems. The amount of sleep deprivation varied over 8, 1, 1!, 0, and " hours without sleep. A total of ten sub#ects participated in the study, two at each deprivation levels. After a specified sleep deprivation period, each sub#ect was administered a set of simple addition problems, and the number of errors was recorded. The followin$ results were obtained% &umber of rrors (y) &umber &umber of *ours +ithout +ithout leep http%--brainmass.com-statistics http%--brainm ass.com-statistics-alltopics-1// -alltopics-1//08 08 *23 *23 +4T* +4T*2T 2T 5 56( 6(*) *) 7 8 8 1 1 1! 1! 0 0 " "
&29: &29:3 3 F 333 333() () 7 ; 8 ! ! 10 8 1" 1" 1 1! 1
.
*our *ourss wit witho hout ut slee sleepp ii. ii. &umb &umber er of erro errors rs
NUMBER OF ERRORS(E) = Y 18 16 14
NUMBER OF ERRORS(E) = Y
12 10
Linear (NUMBER OF ERRORS(E) = Y)
8 6 4 2 0 6
8 10 12 14 16 18 20 22 24 26
MAT530 – 2012, Assoc. Prof. Dr. Adibah Adibah Shuib
?. :ased on the data and variables defines, a)
usin$ a linear e@uation established usin$ the first two coordinates of the data. 7 F(*) 7 a b(*) 8 7 a 8b (1) ! 7 a 8b () a70,b70 7 F(*) 7 0
ii)
usin$ the least s@uare criterion to determine the line of best fit or re$ression line.
*23 +4T*2T 56 (*) 8
&29:3 F 333 () 8
*
*B
!"
!"
!"
8
!
!"
"8
?!
1
!
1""
?!
1
10
1""
10
100
1!
8
/!
18
!"
1!
1"
/!
"
1C!
0
1"
"00
80
1C!
0
1
"00
"0
1""
"
1!
/!
?8"
/!
"
1
/!
88
1""
1!0
10!
880
18"8
1?!
b 7 D n E *B E BE * - D n E * > (E *) 7 D10 (18"8) > 10! (1!0) - D10 (880) > (1!0) 7 1/0 - ?00 7 0."/ a 7 ( E > bB E * ) - n 7 ( 10! > 0."/ B 1!0 ) - 10 7 ?0 - 10 7? 7 F(*) 7 ? 0."/*
MAT530 – 2012, Assoc. Prof. Dr. Adibah Shuib
b) Gompare the e@uations obtained in ?. a) with those found usin$ the 54&T and T3&<54& functions of Hcel. Hplain.
NUMBER OF ERRORS(E) = Y 18 16 14
NUMBER OF ERRORS(E) = Y
f(x) = 0.48x + 3 R = 0.64
12 10
Linear (NUMBER OF ERRORS(E) = Y)
8 6 4 2 0 6
8 10 12 14 16 18 20 22 24 26
2sin$ T3&<54& is more accurate rather than usin$ 54&T function . c) Find the correlation coefficients of all linear e@uations obtained in a) and b). Are these coefficients considered si$nificantI +hy or why notI 4s there a stron$ positive correlation, wea= ne$ative correlation, stron$ ne$ative correlation or no correlation between the two variablesI r 7 D n ( E * ) > ( E * )( E ) - D J ( n ( E * ) > ( E * ) ) B J ( n ( E ) > ( E ) ) 7 D 10 (18"8) > (1!0 )( 10! ) - D J ( 10 ( 880 ) > ( 1!0 ) ) B J ( 10 ( 1?! ) > ( 10! ) ) 7 ( 1/0 ) - ( /!./ B ??./? ) 7 0.801 ;es, these coefficients considered si$nificant because there is no absolute number $uide for correlation coefficient that tell when a two variables have low to hi$h de$ree of correlation. *owever, r closed to 1 or 1 su$$est a hi$h de$ree of correlation, values closed to 0 su$$ests no correlation or low correlation and values between 0. and 0.8 are moderate.
MAT530 – 2012, Assoc. Prof. Dr. Adibah Shuib
".
63<4GT< KA52 ( L ) y 7 0."/H ? !.8 !.8 8. 8. 10.! 10.! 1./ 1./ 1"." 1"." 10!
34<2A 5 ()(L) 1. 0.8 . 1.? .! ?." 1./ 0./ 1.! ." 0
<K4AT4& <K4AT4& ( M ) !.! 1.1! 1.1! 0.?! !.! 11./! 11./! 1.C! C.1! 1.C! 11."
65A4&< <K4AT4& ( L M ) 1"."" 1"."" ?.!1 ?.!1 0 0 ?.!1 ?.!1 1"."" 1"."" .
3 7 D E ( M ) - D E ( L M ) 7 (11.") - (.) 7 1.//! /. +hat is the slope of the least s@uares (bestfit) lineI 4nterpret the slope. i)
lope 7 0."/ n avera$e, for any hour increase in sleep deprivation, a studentNs error incerase by 0."/ •
ii)
4ntercept 7 ? •
n avera$e , a student who #ust wor= up from sleep is eHpected to ma=e ? errors
!. Are there any outliers in the above dataI There is no outlier in the above data.
MAT530 – 2012, Assoc. Prof. Dr. Adibah Shuib
NON-LINEAR ESTIMATES A: NON-AUTONOMOUS DISCRETE MALTHUSIAN GROWTH MODEL
1. Find a /0year population data of a country (preferably between 1C!0 010). 6opulation for 9alaysia from 1C!0 > 010 ;ear 6opulation
1C!0 8,1"0,"0/
1C0 10,8/,/10
1C80 1?,!?,""0
1CC0 1,8"/,?0
000 ,CC,180
010 ,/!/,81
http://www.nationmaster.com/graph/peo_pop-people-population&date=1960
a) Find the population $rowth rate for every 10year period. 6opulation $rowth rate 7 D ( 6opulation present > 6opulation past ) - 6opulation past H 100 O i.
6opulation $rowth rate for 1C0
ii.
6opulation $rowth rate for 1C80 100 O
7 D (10,8/,/10 > 8,1"0,"0/) - 8,1"0,"0/ H 100 O 7 ??.? O 7 D (1?,!?,""0 > 10,8/,/10) - 10,8/,/10 H 7 !.8 O
iii.
6opulation $rowth rate for 1CC0
7 D (1,8"/,?0 > 1?,!?,""0) - 1?,!?,""0 H 100 O 7 C.!! O
iv.
6opulation $rowth rate for 000
7 D (,CC,180 > 1,8"/,?0) - 1,8"/,?0 H 100 O 7 8.8 O
v.
6opulation $rowth rate for 010
7 D (,/!/,81 > ,CC,180) - ,CC,180 H 100 O 7 1C.8O
b) stimate the population and the percenta$e of relative error by usin$ i. The avera$e first four $rowth rates. 6lot the $raph and write the e@uation usin$ best fit curve. Avera$e first four $rowth rates ;A3 1C!0 1C0 1C80 1CC0 000
7 ( ??.?O !.8 O C.!! O 8.8 O ) - " 7 C.!O P3+T* 3AT 0 0.??? 0.!8 0.C!! 0.88
MAT530 – 2012, Assoc. Prof. Dr. Adibah Shuib
Growth Rate 0.3" 0.3
f(x) = 0.01x $ 10.4#
0.2"
%r&' Rae
0.2
Linear (%r&' Rae)
0.1" 0.1 0.0" 0 1!"0 1!60 1!#0 1!80 1!!0 2000 2010
ii. The avera$e of all $rowth rates. 6lot the $raph and write the e@uation usin$ best fit curve. ;A3 1C!0 1C0 1C80 1CC0 000 010
P3+T* 3AT 0 0.??? 0.!8 0.C!! 0.88 0.1C8
Growth Rate 0.3" 0.3 0.2"
f(x) = 0x $ 4.81 %r&' Rae
0.2
Linear (%r&' Rae)
0.1" 0.1 0.0" 0 1!"01!601!#01!801!!0200020102020
MAT530 – 2012, Assoc. Prof. Dr. Adibah Shuib
c) 6lot the $raph of the $rowth rates versus year. stimate the linear model for the $rowth rate as a function of time in year. 2sin$ the linear model for the $rowth rate, find the non-autonomous discete Ma!thusian "o#th mode!. stimate the population and the relative error.
Growth Rate 0.3" 0.3 f(x) = 0x $ 4.81
0.2"
%r&' Rae 0.2
Linear (%r&' Rae)
0.1" 0.1 0.0" 0 1!"01!601!#01!801!!0200020102020
6n1 7 f ( tn , 6n ) Q 60 7 8,1"0,"0/ Q k(t) = 0.0025 t - 4.8076 6n1 7 ( 1 = ( t n )) 6n 6n1 7 ( 1 ( 0.0025 t - 4.8076 )) 6n 6n1 7 ( 0.0025 t - 3.8076 )) 6n
;A3 1C!0 1C0 1C80 1CC0 000 010
P3+T* 3AT (y) 0 0.??? 0.!8 0.C!! 0.88 0.1C8 1.?8/"
T49AT< KA52 (R) y 7 0.00/ t ?.80! 1.0C" 1.11" 1.1"" 1.1!" 1.1C" 1.1" !.CC"
333 (y R) 1.0C" 0.8" 0.8" 0.808 0.C0? 1.018 /./""
MAT530 – 2012, Assoc. Prof. Dr. Adibah Shuib
d) Gompare $raphically the population from the models. stimate the population of the country in 0/0 usin$ estimates found in b) and the Nonautonomous Ma!thusian Go#th mode!$ 1. 6n1 7 ( 1 r ) 6n Q 60 7 8,1"0,"0/ Q r 7 0.08 6n 7 ( 1 r )n 60 6C 7 ( 1 0.08 ) C (8,1"0,"0/) 7 ?,//",""8.!8 S ?,//",""C
Population 30*000*000 2"*000*000
f(x) = 3!326".##x $ #63##1#68.62
20*000*000
&,-ai&n Linear (&,-ai&n)
1"*000*000 10*000*000 "*000*000 0 1!"01!601!#01!801!!0200020102020
.
6n1 7 ( 1 = ( t ) ) 6n 6n1 7 ( 1 ( 393266 n - 800,000,000 ))6n 6n1 7 (393266 n - 799,999,999 )6n
MAT530 – 2012, Assoc. Prof. Dr. Adibah Shuib
%: LOGISTIC GROWTH MODEL
The followin$ data was collected from an eHperiment measurin$ the $rowth of a yeast culture
5et
Time&hou'
(east %iomass
Time&hou'
(east %iomass
1 ? " / ! 8 C
10 18 C " 1 11C 1/ / ?/1
10 11 1 1? 1" 1/ 1! 1 18
""1 /1? /!0 /C/ !C !"1 !/1 !/! !!0
P n be
the yeast biomass at the end of n hours.
a) 6lot the $raph of P n1 versus P n. stimate the polynomial of de$ree to fit the data and use this polynomial to obtain a lo$istic $rowth model.
Yeast Biomass #00
f(x) f(x)==4#.86x $ 0.#3x/2 $ !#.84 + 61.#!x $ 144.28
600 Yea Bi&a
"00 400
&n&ia (Yea Bi&a)
300
Linear (Yea Bi&a)
200 100 0 0
2
4
6
8 10 12 14 16 18 20
5o$istic $rowth model is Pn+1 = -0.7332 ( Pn )2 + 61.791Pn - 144.28
MAT530 – 2012, Assoc. Prof. Dr. Adibah Shuib
b) From the $raph of population versus time, the population appears to be approachin$ a limitin$ value or carryin$ capacity. Puess a suitable value for the carryin$ capacity M . 2sin$ the value M and the lo$istic $rowth model P n17 P nk ( M P n) P n, estimate the value of k . Pn+1 = -0.7332(Pn)2 + 61.791Pn - 144.28 Pn+1 = ( 1 + r ) Pn 1 + r = 61.791 ; r = 60.791 r / M = -0.7332 ; M = 60.791 / -0.7332 = -82.91 ( Carrying capacity )
6n1 7 6n = (-82.91 6n ) 6n 61 7 60 = ( -82.91 60 ) 60 18 7 10 = ( 8.C1 > 10 ) 10 18 7 10 > CC.1= CC.1= 7 8 = 7 8 - CC.1 = 7 0.008!1
c) 6lot the $raph of the yeast biomass versus hours for the two models and the observation.
Yeast Biomass #00 f(x) f(x)==4#.86x $ 0.#3x/2 $ !#.84 + 61.#!x $ 144.28 600
Yea Bi&a
"00 400
&n&ia (Yea Bi&a)
300
Linear (Yea Bi&a)
200 100 0 0
2
4
6
8
10 12 14 16 18 20
MAT530 – 2012, Assoc. Prof. Dr. Adibah Shuib
T49(*23 ) 1 ? " / ! 8 C 10 11 1 1? 1" 1/ 1! 1 18
•
;AT :49A (y) 10 18 C " 1 11C 1/ / ?/1 ""1 /1? /!0 /C/ !C !"1 !/1 !/! !!0 !"?
63<4GT< KA52 R 7 ".8!1H C.8"? "C.C8 .11 "/." C?.!01 1"1."! 18C.?? ?.18" 8/.0"/ ??.C0! ?80.! "8.!8 "!."8C /".?/ /.11 !0.0 !!.C?? 1/.C" !?.!// !"?.0/
333 ( y R) ?/C.8"0?" "0".8/"!"1 80.! 11.!/?01 "C!".8C?""" "C"/.?"?C ?8!!.8"C8/! 8!./0/ ?.?C8?! ?!8.01"8C 118.!?"?8" !C".0811 "CC1."/ ?".CC0/1 "?.C8118" 8!.!"8C ?//.?"?! 10"".?/C0? !?.0C!1
um of s@uares of errors is !?.0C!1
MAT530 – 2012, Assoc. Prof. Dr. Adibah Shuib
