Question

A certain manufacturing firm produces a product that is packaged under two brand names, for marketing purposes. These two brands serve as strata for estimating potential sales volume for the next quarter. A SRS of customers for each brand is contacted and asked to provide a potential sale figure y in number of units) for the coming quarter. Last year's true sales figure, for the same quarter, is available for each of the sampled customers and is denoted by x. The data are given in the accompanying table. The sample for brand I was taken from a list of 120 customers for whom the total sales in the same quater of last year was 24,500 units. The brand II sample come from 180 customers with a total quarterly sales last year of 21 200 units. Brand I Brand II Ii 204 143 82 137 189 119 150 200 125 210 160 75 280 300 190 256 60 275 198 103 107 159 110 100 180 75 63 87

Answer 1

Answer:

Given data

A certain manufacturing firm produces a product that is packaged under 2 brand

The sample for brand 1 was taken from a list of 120 customers for who the total sales in the same quater of last year was 24 units

the brand 2 sample come from 180customers with a total quarterly sales last year of 21200 units

x i	y i	(y - $\gamma$ X)2
204	210	16.33332
143	140	99.22431
82	75	121.7992
256	280	129.9364
275	300	131.4171
198	190	314.9246
total = 1158	total = 1215	total = 813.6349

Now let T s  = total sales

= 24,500

$\bar{y}$    = $\frac{1}{n}\sum_{i=1}^{n}y_{i}$

=   

(1215)

= 202.5

$\bar{x}$ = $\frac{1}{n}\sum_{i=1}^{n}x_{i}$

=

(1158

= 193

Let ratio etimator $\gamma$ = $\frac{\bar{y}}{\bar{x}}$

= $\frac{\bar{202.5}}{\bar{193}}$

= 1.049223

This total ratio estimate

barT,=
  $\frac{\bar{y}}{\bar{x}}$ Tx

= $\gamma$ Tx

= 1.049223 * 24500

= 25705.96

Let $S_{r}^{2}$ =  

– ፡- 7( EC - 6)

=   

(813.6349)

= 162.727

Variance $(\bar{T_{r}})$ = $\bar{N}$
(n - n)
$\frac{S_{r}^{2}}{n}$

=  $\frac{T_{r}}{\pi }(\frac{T_{r}}{\pi }-1)\frac{S_{r}^{2}}{n}$

=   

162.727 | 24500、24500 ( 109 八 103 - ||

= 433581200.7

Let SD

(Tý) = F433581200.7

= 20822.612

Brand 2

xi	yi	(Y - $\gamma$ x)2
137	150	21.122392
189	200	0.352792
119	125	1.6897447
63	60	47.12348
103	110	0.464574
107	100	183.9761
159	180	126.4804
63	75	66.18385
87	90	5.461118
Total = 1027	1090	452.8544

Let Tr = Total sales = 21200

  

16012 = 3 1

= 121.111

i = 1027

= 114.111

Let estimator $\gamma = \frac{\bar{y}}{\bar{x}}$

  

121.11 114.111

7= 1.061344

$\bar{T}_{r}= \frac{\bar{y}}{x}T_{r}$

= 71, = 1.061344 x 21200

= 22500.49

where

s - Σκι - γX)

= (452.8544)

= 50.32

Let Var

(T) = ÑÑ - n). Se

一日

  

50.32 21200721200 114.111114.111

= 191938.53

where

SD(T) = V191938.53

= 438.11

By solving both Brand (1) & Brand (2)

SD Brand 2 < SD Brand 1

Therefore Brand 2 data is the best data.