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CGSf fOIOTlOMS HR SAiRPU SOETIfS
'NpvMI ■pBhaene Hr ■Pw "*e PP®w P *tt>He ■*■
Saraet Ernest McCreary
.4 Meeertatioa Safcaitted to the
SradMate Faculty la Fartial Fulfillment of
the !§<ptf«w#ate for the Degree «f
soofos o? mmmm
»S#er Safcjeet % Statistics
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Iowa State College
1950
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UMI Number: DP12481
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Page
PAlf 1, tSnt@0QWI©V.......................................... I
?jm ii. nrxw ©f ittm m s ......................................................... 3
A. fm vel functions ami. field Costs ............... 3
1. Decision Tanetieao. ............ 5
nun III. SPSSf® MIK0CW T
A. Palate eat Sampling ©nit*................................ I
S. fbe Theoretical Prolsles of Miaisua Distance... f
C. Grid. art. Airline Bietance then a * 2................ I
9. Mean Sgaare Bietaac© and Ixpeetod Grid
Distance far a * 2 Whoa the Palate Save a«jr
Space Distribution orer any Plane Area 11
m
X. liata ©fid Bietance Anoag a • J Palate
la a S fu ero ........................................................ 13
7. Impirical tarestigations of Mininaa
Arerage Dletaacee............................................... IS
ft. Aa Tipper Bound te Miataaa Grid Distance
in a Xeetaagle. .......................................... 23
mm $
t u t if. ix iA fm s * ©f t© A tn as »ss»juws 2
FAW ▼® w. arewB.M.. BB De.I sS. TAUC® Oas*V ™ W -Ams •SwSe™SI eCe SF .f©mtm. A
PUBIS IGSAfIB BIKECf I0»................................. 35
A. irl# Bietance Among a Polnte la a lettaagle... 35
5. Grid Distance Aaong a Point a IStea a
lectangle is Stratified.. ........... A©
©. Mean Sfuare Bietanee Among n Points la a
Circle, Seaiciral# and Quadrant..................
B. Grid Mstance frea a fia d Point t© a
land©* Point............. $$
PAW ft. ASALY5IS Of TUB A® KXWftSK I» «® W *» fOX
vnas iowa faxm sm fsti.*... ............. 72
'"A. Description of the Smrfojre. ......... f§
X. Stapling Wait Mileage................................ #S
6. See* Mileage.. ............... 9®
B. least ffmnres Settaction ef the teaataiag
Mileage. ........ $f
7 ? * 3 (
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lit
PAlf VII. HfMMISHe SAMPLE SIZS 17 USE Of fHI
mifijttx m m m m ..........................................
..............
wm ?iii. sitCfsiwr ms mmmt.................. 109
piBff 11. mwnmmm................. n€
vm 1. x isw a n s m s». ......... 111
pass xi. Afinwxsswnr.............................................. i2i
fA » XII. .IPfl81II». ,...---------- 129
A. Investig&tlea «# the Slope of 2{t) foe •
0 < k < 1............ 126
1. f OHIO foot. ........................ 130
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PART I. IWlOOTCflOS
la planning sample surreys administrators are interest#! in both
the accuracy and the ©sat'of various designs. Usually there are many
possible sampling designs and sices of samples which will glwe the re-
faired accuracy but the corresponding costs are unknown.
Any overall cost function should include office costs as well as
field costs. Phis study concerns itself with field costs which are by
far the more difficult to estimate, The field costs considered, are
those for survey* in which the investigators east travel to certain
random locations in a geographical area. Under these circumstances,
if the total distance travelled and its various component* can be esti
mated from a travel function, if the speed in m.p.h. for each component
of travel can be prognosticated, and if the average time to make an
observation or interview I* known, it ie possible to arrive at a cost
function for the field work.
In this thesis,theory is presented that will enable on# to make an
estimate of the total distance travelled in a survey for a wide variety
of sample designs where the population has a uniform space distribution
over any stratum. Interviewers kept mileage and time records for three
different Iowa farm surveys, The data derived from these records are
presented and compared with theoretical results.
fhe overall problem of deciding what is the accuracy refuired is
much more difficult and brings up the question of the loss that is
sustained from errors la estimates, as well as the cost of the survey.
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2
la fart TII toate of the elementary notions of laid'# decision fanction
are applied to determining sample else, where ©me does mot know the
parameter# of hi# distribution of error.
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3
MU IX. » ? iw Of M fltifW #
A. fra##! Functions aad field Gotti
la t#*i function® began la Ifjh wh*a leyasai (11) showed
there wee mm opiiauss rat® of sampling for each strati** depending ©a the
rariation vlt&ia and the sis# of each strata*. Dfclftg lagraaglft* m itt*
pliers,, It li possible to deter*t*e the opt tana rate# within a particular
design If oat ftla-lai*## «o«t for a fixed accuracy or alternately *axi-
alses tho eeearaeir f»* » fixed ooot* lot## m i Mmmpmw '01) la 1935
described a u iM of ietewlftinf the eftlaal percentage of sampling
tm field oapirlftoftt* taking iat# account the cost of harvesting serlouss
els# saaples. They mlm ixveetlgated the opt tea* else of the eaapllag
ftttit la a particular instance.
P. 0, Mkhftlaftfthlft (l%) la eonsid#red * cost fmaotloa which
depended partially «ft th# distance tme®lied, li stated that it vm
easy to eee that the expected length of a remt# ooaaoetiaf a point#
scattered at randea within any given are* *mm
la 19&2 lessen (ill «*ei this a# a hail# for setting up * field cost
function for a fa** surrey la Iowa. Mahalaaobis and dessea in th#ir
respective pipers set mp a variance f aaetion showing hew the saapling
nnMLftaeblf should has# stated that it equals \fi SLzJk fer a
systeeatie ftmpling in a plane Cse* tart f.A). \pm
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%
error varied with the else of the sampling wait Cor with mean square
distance with1a an s.u.). It was possible then to arrive at two equations
using a Lagrange multiplier and thus to obtain formally, for a given
total cost, the optimum sis® and number of sampling units for maximum
accuracy.
Several research workers Cincluding Mafaalanobls) realized that
expression (1) is not the exact expected minimum distance among n random
points. 1. Ghosh (7) In 19^5 obtained the distribution of the distance
between'two points where each point had a uniform distribution In the
rectangle ab. Previous to this, Williamson's “Integral Calculus* (J©)
had given the mean (the expected distance) of this distribution and
Jesses (11) had shown the uncorrected second moment. Independent of
1 1
Ghosh and independent of each other, A. M. loot and 1. G. S. The* in
19^5 also found the distribution of the distance between two random
points In a rectangle.
g
Marks (1$) in Ifhf obtained a lower bound for the expected travel
in any bounded two dimensional lo rd set of area A to be
| U . (2)
VaT
An upper bound of the expected minimum distance was found by *?• H» Ghosh
1
Both solutions are unpublished but Mood's derivation may be found
in the July Ifbf Progress Beport to B.A.X. Ames, Iowa, Statistical
laboratory, pp. hhJig. See reference (17b).
g.
fhis was Misprinted at the beginning of larks* not®, but was given
correctly at the end of the note. M. S. §ho«h (7) used the incorrect
expression. ■: ? i
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