2011 National Survey of Fishing, Hunting, and ... - Census Bureau

Illustration of the Computation of the Standard Error of an Estimated Percentage
Suppose there were an estimated 13,674,000 hunters age 16 years old and older of whom 18.9 percent hunted migratory
birds. From table D-2, the appropriate b parameter is 15,798. Using formula (3), the approximate standard error on the estimate
of 18.9 percent is
5, 756 18. 3 100 , .
18.
3
,
5 5756 , 756 1818 3. 3 100 100
1818 . 3
.
3083
.
s xp s , xp ,
12, 510,
000
083
. 083 .
12 12 , 510 , 510 , 000 , 000
s xp
Consequently, the 95-percent confidence interval for the estimate percentage of migratory bird hunters 16 years old and older
Consequently,
is from Consequently, 16.7
the
percent
95-percent the the to 19.9 95-percent confidence
percent, confidence interval
ie. 18.3
for interval
± interval 1.96
the estimate
x
for
0.83. for the the percentage estimate estimate percentage of migratory of of bird migratory hunters bird bird 16 hunters hunters years 16 old 16 years and years older
old and and older olde
is from 16.3 is is from percent from 16.7 16.7 to percent 21.5 percent percent, to to 19.9 19.9 i.e., percent, percent, 18.9 ie. ± ie. 1.96 18.3 18.3 x ± 1.33. 1.96 ± 1.96 x 0.83. x 0.83.
Standard Error of a Difference. The standard error of the difference between two sample estimates is approximately equal to
Standard Error Standard of a Error Difference. Error of of a a Difference. The standard The The error standard standard of the error error difference of of the the between difference two between between sample two two estimates sample sample is estimates approximately is is approximately equal to equal equal to to
2 2
sxsy s
x y x2s2 sxy
2 2
(4)
s s
(4) (4)
x y xysy
(4)
where
where s x
and where sx
and
s y
are sx
and sy
are the standard errors of the estimates x and y. The estimates can be numbers, percentages, ratios, etc. This
will represent where the sy
standard are the standard errors of errors the estimates of the x and y. The x and estimates y. The can be numbers, can percentages, ratios, etc. ratios, This etc. This
sx
and the actual sy
are the standard standard error errors quite of accurately the estimates for the x and difference y. The estimates between estimates can be numbers, of the same percentages, characteristic ratios, in etc. two This
will represent
different
will will the
areas, represent actual
or
the
for
standard the actual
the actual difference
standard error standard quite
between
error accurately error quite quite separate for accurately the
and
difference
uncorrelated
for for the the between difference characteristics
estimates between between in
of estimates the
the
same
same of of area.
the characteristic the same
However, same characteristic in
if
two
there
in in two two
different
a high
different areas, different positive
or areas, for areas, (negative)
the or difference or for for the
correlation the between difference between
separate between between the
and separate
two separate uncorrelated
characteristics,
and and uncorrelated characteristics
the formula characteristics in
will
the
overestimate
same in in the area. the same same However,
(underestimate)
area. area. if However, there is
the
if
true if there there is is
a high
standard
positive a high a high error.
(negative) positive positive correlation (negative) correlation between between the between two characteristics, the the two two characteristics, the formula the the will formula formula overestimate will will overestimate (underestimate) (underestimate) the true the the true true
standard standard error. standard error. error.
Illustration of the Computation of the Standard Error of a Difference
Illustration Illustration of the Computation of of the the Computation of the Standard of of the the Error Standard of Error a Error Difference of of a a Difference
In Table 8, of the 11,655,000 females in the age range of 18-24, 726,000 or 6.2 percent are sportspersons. Similarly, of the
Suppose
11,638,000
In there In Table Table were
males
8, 8, an of of the estimated
in the the 11,655,000 same
13,608,000
age
females
range, females females
1,929,000
in in the the age in age the range
or range age
16.6
of range
percent of 18-24, 18-24, of 18-24
are
726,000 726,000 sportspersons.
of whom or 6.2 6.2 percent 726,000 percent The
are
apparent
or are 5.3 sportspersons. percent
difference
were
between Similarly, of of the the
sportspersons.
the percent 11,638,000 Similarly,
of female
males males suppose in
and in the
male the same there same participants
age were age range, range, estimated
is 10.4 1,929,000 percent.
12,909,000 or or 16.6 16.6 Using
percent males percent formula
in are are the (3) sportspersons. same
and
age
the appropriate
range The The of apparent whom apparent b parameter
2,160,000 difference from
or between between table
16.7 percent
D-7,
the
the percent were
approximate percent sportspersons. of of female female standard
and The male
errors male apparent participants of 6.2
difference
percent
is is 10.4 10.4 and
between percent.
16.6 percent. the
percent
Using percentage Using are
formula formula 0.55
of
and
female (3) (3) 0.85,
and and and the
respectively. the male appropriate sportspersons
Using
b b parameter formula
is 11.4 from
(4), from table
the table
percent.
approximate
Using D-7, D-7, the formula the standard approximate (3) and
error
standard the standard of
appropriate
the estimated
errors errors of b of parameter 6.2
difference 6.2 percent percent from
of
and and 10.4
table 16.6 16.6 percent
percent D-2, percent the
is
are approximate are 0.55 0.55 and and 0.85, standard 0.85, respectively. errors of Using 5.3 Using percent formula formula (4), (4), the the
and 16.7 percent approximate are 0.79 standard standard 1.35, error error respectively. of of the the estimated Using difference formula of (4), of 10.4 the 10.4 percent approximate percent is is standard error of the estimated difference
of 11.4 percent is
2 2
s x
s y
055 . 20 2
x s xy
y
055 . 2. 85 102
055 0 . 85
2.
. 0. 85 102
. 102 .
The 95-percent confidence interval on the difference between 18-24 year old female and male sportspersons is from 8.4 to
The 95-percent
12.4,
The The i.e., 10.4 95-percent confidence
± 1.96 confidence interval
x 1.02. Since
interval interval the
the
difference on
interval on the the between difference does not
18- between
contain between to 24-year-old
zero,
18-24 18-24 we
year year can
female old
conclude old female and female male
with
and and male sportspersons
95 male percent sportspersons confidence
is from is is from 8.3
that from 8.4
the 8.4 to to
to 14.5,
percentage
i.e., 12.4, 12.4, 11.4 i.e., i.e., of
± 10.4 1.96
18-24 10.4 ± x 1.96 ± year
1.56. 1.96 x
old
Since 1.02. x 1.02. female
the Since Since interval the interval does not does contain not contain zero, we zero, can we conclude can with 95 with percent 95 percent confidence that the that the
percentage of 18- to of 24-year-old 18-24 year female old 15.8 sportspersons the interval
female sportspersons to 17.6, i.e., is 16.7 is less does
less is than ± than not
less 1.96 the contain
the than x percentage 0.45. percentage zero, we of can 18-24 conclude year old with male 95 sportspersons.
percent confidence that the
percentage of 18-24 year old female sportspersons is less than the the percentage of 18- of to of 18-24 24-year-old 18-24 old male old male male sportspersons.
Standard Errors of Estimated Averages. Certain mean values for sportspersons, anglers, etc., shown in the report were calculated
Errors
Standard
as Standard the
of
ratio
Errors Estimated Errors of two
of of numbers.
Averages. Estimated For
Certain Averages. example,
mean Certain Certain average
values mean mean for
days
values sportspersons, values per
for
angler for sportspersons, is
anglers,
calculated
etc., anglers,
as: anglers, shown etc., etc., in shown the shown report in in the were the report report calculated
as the lated lated ratio as as of the the two ratio ratio numbers. of of two two For numbers. example, For For average example, days average average per angler days days per is per calculated angler angler is is as: calculated as:
were were calcu-
calcu
as:
x
y
total days
x
x total days
total days
total anglers
y y total total anglers
Standard errors for these averages may be approximated by the use of formula (5) below.
Standard Standard errors Standard for errors these errors for averages for these these averages may averages be approximated may may be be approximated by the use by of by the formula the use use of of (5) formula formula below. (5) (5) below. below.
x s
2 s
x s
s
s
r ss
x y x y
x y
(5) (5)
y
x
2 s
s
s
y
2 r ss
x y x y
x y
(5)
y
x
2
s
r ss
x y x y
x y
y
(5)
y
x
y
2
xy2
xy xy
In formula In formula (5), r (5), represents r represents the correlation the correlation coefficient coefficient between between the numerator the numerator and the and denominator the denominator of the of estimate. the estimate. In the In the
above formula, In formula use 0.7 (5), as r an estimate the of r. 15.8 to 17.6, i.e., 16.7 between ± 1.96 the x 0.45.
above In formula, use (5), 0.7 r represents as an estimate the correlation of r. coefficient between the numerator and and the the denominator of of the the estimate. In In the the
above above formula, formula, use use 0.7 0.7 as as an an estimate estimate of of r. r.
Illustration Illustration of the of Computation the Computation of the of Standard the Standard Error Error of an of Estimated an Estimated Average Average
Illustration of of the the Computation of of the the Standard Error Error of of an an Estimated Average Average
Suppose Table that 2 the shows estimated that the number average of days the per average angler days 16 per years angler old and age older 16 years for all old fishing and older was for 17.3 all days. fishing Using was 16.7 formulas days. (1) and
Using (2) formulas Table
above, Table 2 (1) we
shows 2 shows and compute
that (2) that above, the
the the average
standard average we compute days
error days per
on the per angler
total standard angler days,
16 16 years error years 516,781,000,
old on old total and and older days, and older for
total 553,841,000, for all
anglers, all fishing fishing 29,952,000,
was and was 17.3 total 17.3 days. anglers, days. to be
Using
15,828,079 Using 33,112,000, formulas formulas and to (1) (1) and and
be 20,329,124 399,342,
(2) (2) above, above, and respectively. 693,033, we we compute compute respectively. The
the
approximate the standard standard The error approximate standard error on on total
error total days, standard on days, the 516,781,000, estimated error on the average
and estimated and total total of
anglers,
17.3 anglers, average days 29,952,000, is of 16.7 days to to isbe 15,828,079 and and
399,342, 399,342, respectively. The The approximate standard standard error error on on the the estimated average average of of 17.3 17.3 days days is is
2 2
2 2
516, 781,
000 158, 280,
079 2 2
516 781 000
s xy
2
, , 158 , 280,
079
399,
342
15, 828, 079 399,
342
s xy
2
07 .
0 40
29, 952,
000 516
.
, 781,
000 s xy
516, 781,
000 158 280,
079
29 399,
342
15, 828, 079 399,
342
399,
342
15, 828, 079 399,
342
2
07 .
0.
40
29, 952,
000 516, 781,
000 29, 952,
000
, 952,
000
2516 07 . , 781, 000 29, 952,
000 0.
40
29 952,
000 516, 781,
000 29 952,
000
516 516 , 781 , 781 , 000 , 000
2929 , 952 952 , 000 , 000
Therefore, the 95-percent confidence interval on the estimated average of 16.7 days is from
15.8 to Therefore, 17.6, i.e., the 16.7 95-percent ± 1.96 x 0.45. confidence interval on the estimated average of 17.3 days is from
16.5 to Therefore, 18.0, i.e.,
the the 17.3 95-percent ± 1.96 x 0.40. confidence interval interval on on the the estimated average average of of 17.3 17.3 days days is is from from
16.5 16.5 to to 18.0, 18.0, i.e., i.e., 17.3 17.3 ± 1.96 ± 1.96 x 0.40. x 0.40.
U.S. Fish and Wildlife Service and U.S. Census Bureau 2011 National Survey of Fishing, Hunting, and Wildlife-Associated Recreation—Vermont 69
15.8 to 17.6, i.e., 16.7 ± 1.96 x 0.45.