ratio estimation(edited)

8/13/2019 Ratio Estimation(Edited)

http://slidepdf.com/reader/full/ratio-estimationedited 1/21

1



• Technology Oriented • Business Driven • Sustainable Development • Environmental Friendly

Note

To accomplish the sampling in (iv), take a random sample of n units; then take a pair of(x,y) measurements on each unit.

Problem: Make statistical inferences for

a) R : POP ratio =

b ‘y’ mean

c) Y (or T ) = POP ‘y’ total

Definition. The Ratio Estimate for

(a) R is:

(b) is :

(c)Y (or T) is :

= =

X Y

X Y

POP Y

x y

x

yr

Y X x

y X r y

r

X Nr y N r

X x

y N rX

3




Definition:

Where

Definition: (Statistical Inference: RE)

1)POP para : R= POP ratio

Sample Estimate = r= y/x =

Std. Error: , % CI :

n

i

iir

n

rx y s

1

2

2

1

1

2 222

n

x sr xyrs y s

i

n

i

i

n

i

i

n

i

i

y x xy s

x x s

y y s

1

2

1

2

1

22

/ y x

2

2

1ˆ r S

X n

f r e s 100(1 ) r e s Z R ˆ

2 4




2) POP para : =POP mean

Sample estimate :

100 (1- ) % CI:

3) POP para: Y=POP total

sample estimate :

std. error:

100

Y

r

X y r X r

N

y X

x

21

ˆ r r

f se y S

n

2

ˆ r r y Z se y

r Ny NrX rX

ˆ ˆr r se Ny N se y

21r

f N S

n

r r y N e s Z y N CI ˆ:)%1(

2

5




Example

The POP ‘x’ of 69 urban areas of Greece are known from a previous census in 1941, the total

POP being X = 1,355,900. Twenty cities are selected at random and their present 1951 ‘y’

POP determined.

City X Y City X Y 1 12,200 12,600 11 15,600 22,400

2 29,000 25,700 12 32,300 41,000

3 7,000 10,800 13 15,300 13,100

4 14,100 18,500 14 30,400 29,500

5 49,700 42,100 15 13,900 17,700

6 13,000 14,800 16 62,300 79,000

7 8,700 11,900 17 15,000 17,600

8 19,800 33,800 18 21,700 23,600

9 34,700 36,800 19 9,100 12,900

10 15,100 22,100 20 8,300 12,100

6




N=69

n=20

s (x) =427,600

s (x 2 ) =13,169,560,000

651,1969

900,355,1 x

000,500 y s

000,140,411,172 y s

000,320,883,14 xy s

600,427

000,500

x yr

169.1

1

2 2222

n

x sr xyrs y sS r

000,158,3219

000,000,611

7




Goal:

i) Estimate the rate of growth

ii) Estimate the present POP Y

iii) Find a 95% CI for Y

Solution:

i) = Estimate for rate of growth

ii) = 69(1.169)(19.651)

= 1,585,069.311

iii)

∴ 95% CI for Y is 1,585,069,311 ± 1.96 (73731.34)

1,585,069.311 ± 144,513.42

169.1ˆ

r R

r Ny NrX

34.73731

000,158,3220

69/201

69

1ˆ

2

r r S

n

f N y N e s

8




Note:

When are the ratio estimates preferable to the single sample estimates and for

and respectively?

Answer:

Let = the POP correlation coefficient of ‘x’ and ‘y’

=

i) If then the ratio estimates are preferable

ii) If CV(x) = CV (y), then the ratio estimates are preferable if

Example

For the POP survey of Grecian cities using ratio estimate, is the ratio estimate preferred to the

total estimate for estimating the total POP of the 69 cities in 1951?

y y N Y

Y

N

i

N

i

iii

N

i

i Y y X xY y X x1 1

22

1

)(/

,/

/

2

1

)(

)(

2

1

Y

X

yCV

xCV

y

x

ye s ye s r ˆˆ

2

1

Ny 9




Solution

An estimate of is = assuming

Hence, we conclude that the ratio estimate is preferred to the estimate

Samples Size Determination in Ratio Sampling

Let be the POP value of .

Assume and are given :

The sample size required to estimate to closeness with confidenceusing the ratio estimate is:

n

i

n

i

ii

n

i

ii

y y x x

y y x x

1 1

22

1ˆ ,

2

1943.

)1951()1941(

)()( yCV xCV

y N

22

1 1

N i i

Ri

Y RX S

N

2

r s

1 ˆ 1 P e

n

e

1ˆ

N

DS

DS n

R

R

2

2

1

e

10




2

2

22

2

2

22

2

2

2

/

ˆ

e

Z N X Nr y N Y

e

Z X r r yY

X e

Z x yr R

D

r

- POP Ratio

- POP Mean

- POP Total

Example (Refer to handout)

Solution

i) The estimate of Y is

Note that: N is not needed to estimate Y

X x

y X

x

y N y N r

.246(1800)

.4.35

101.79 lbs

11




ii)

Assume ~ 1 and since N is unknown we estimate N by

154.310

000005812.

35.4

18000

1ˆ

0566.35.4

246.

000005812.

9

000052285.

9/9035.1.0566.010839.)0566(.2006224.

1

2

2

2

2222

r r

r

sn

f N y N e s

r

n

x sr xyrs y s

s

N n N f 1

35.4

1800

35.4

1800

10 N

N

12




Note

∴ A 95% CI for Y is

101.79 ± (1.96)(3.154)

101.79 ± 6.18 = (95.61, 107.97)

iii)

xnX N

x X

n N

22

2

2 2

2 2

2

2

,

1

.000005812

18000 413.8

4.35

4138 1.960.0000058129

42.48

42.48 42.4842.05

42.48 1.01031

4138

43

r

r

R r

r

D N Z S Dn

S D e

N

S is estimated with s

N is estimates by

s D

n

13




Ratio Estimation in Stratified Random Sampling

For the same reason indicated earlier, it is sometimes advantageous to stratify the POP before

using a ratio estimator. We will assume that we can take a large enough sample of both x’s

and y’s in each stratum for the variance approximation to work fairly well.

Basically, there are two different methods for constructing estimators of a ratio in stratified

sampling:-

1. Estimate the ratio of to within each stratum and then form a weighted average

of these separate estimates as a single estimate of the POP ratio. The result of this is

called a separate ratio estimator.

2. Estimate by the usual and similarly estimate

3. Then can be used as an estimator of This is called a combined ratio

estimator

strs strs X Y

strsY strs y strs strs X by x

strs

strs x y Y X

14




Separate Ratio method of .

Combined Ratio method of .

strsY

L

L

L L RS X

x

y

N

N X

x

y

N

N Y

...ˆ

1

1

11

1

1 1

22

1

2

1

11

11

2

1

1...

1

ˆˆ

n

i

n

i L

i Li

L L

L L Lii RS

l

n

xr y

n N

n N

N

N

n

xr y

n N

n N

N

N Y V

strsY

Ln

i L

ici

L L

L L L

n

i

ic

RC

strs

strs RC

n

xr y

n N

n N

N

N

n

xr y

n N

n N

N

N

Y V

X x

yY

1

22

1 1

2

1

11

11

2

1

1...1

ˆˆ

ˆ

1

15




Example (refer to handout)

i. Find the separate ratio estimate of and its estimated std. error

ii. Find the combined ratio estimate of and its estimated error.

iii. Compare the procedures.

Solution

No. of strata = 2, i.e.

Company A Company B

strsY

strsY

8.1710

178

7.1810

187

300,16

10

0000,1

A

A

A

A

A

x

Y

X

n

N

8.710

78

6.410

46

800,12

10

5000,1

B

B

B

B

B

x

Y

X

n

N

16




i.

and similar calculations for Company B,

B

A

n

i B

i Bi

B B

B B B

n

i A

i Ai

A A

A A A RS

B

B

B B A

A

A A RS

n

xr y

n N

n N

N

N

n

xr y

n N

n N

N

N Y V

X x

y

N

N X

x

y

N

N Y

1

2

1

22

1

1

ˆˆ

87.9

500,1

800,12

8.7

6.4

500,2

500,1

000,1

300,16

8.17

7.18

500,2

000,1

ˆ

265.31

)24.4)(05.1(2)066.4()05.1(463,4

2

2

10

1

10

1

2210

1

10

1

22

iii A

i

ii Ai y xr xr y xr y

17




6325.0ˆ

ˆ

40.09

45.87

)10(1500

10500,1

500,2

500,1

9

26.31

)10(000,1

10000,1

500,2

000,1ˆˆ

45.87

22

210

1

RS

RS

i

i Bi

Y e s

Y V

xr y

ii.

24.1076.248.7

)6.4(500,2

500,1)7.18(

500,2

000,1

ˆ

2

1

B B

A A

h

hh

strs

strs

strs RC

y N

N y

N

N

y N

N y

X x

yY

18




80.1168.412.7

)8.7(500,2

500,1)8.17(

000,2

000,1

B B

A A

strs x

N

N x

n

N x

8678.080.11

24.10

11

ˆˆ

13.10)64.11(

80.11

24.10ˆ

64.112500

800,12300,16

1

22

1

22

strs

strsc

n

i B

ici

B B

B B B

n

i A

ici

A A

A A A RC

Rc

B A

x

yr

n

xr y

n N

n N

N

N

n

xr y

n N

n N

N

N Y V

Y

n

X X X

B A

02.1ˆ

ˆ

04.1ˆˆ

56.192

25.154

1

2

1

2

RC

RC

n

i

ici

n

i

ici

Y e s

Y V

xr y

xr y

B

A

19




iii) Comparison of the separate ratio and combined ratio estimators

Estimation type Estimate Std. error

Separate 9.87 0.63

Combined 10.13 1.02

Note

The variances of each can be adjusted accordingly.

NoteIf the stratum sizes > 20, use separate ratio estimate. If stratum samples sizes are small or if

within- stratum ratios are ≈ equal, then combined ratio estimator may perform better.

RC RC

RS RS RS RS

Y N Y

Y e s N Y N e sY N Y

ˆˆ

ˆˆ

ˆˆ,ˆˆ

20




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ratio estimation(edited)

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