Question

A Canadian grocery store chain has just acquire another chain of grocery stores with different target customers from the first. Before reviewing its new product distribution policy, following this acquisition, the chain of stores appointed an analyst to study the markets represented by urban agglomerations whose populations are located between 23,000 and 110,000 inhabitants. To do this, we analyze the data relating to a sample of 49 branches of the chain. The variables on which the information collected relates are:
• VNT, the branch's annual sales volume in thousands of $ for the previous year;
• POP, the estimated population in thousands of inhabitants of the urban agglomeration served by the branch during the previous year;

• NBM, the number of stores, of the same type as those of the chain studied, in operation for a year in the urban area served by the branch (this number includes the branch).

Branches	VNT	POP	NBM
S1	4 197	89,5	8
S2	4 622	92,6	8
S3	4 721	96,3	8
S4	6 817	98,4	7
S5	6 747	99,5	7
S6	4 830	25,4	2
S7	3 490	27,3	3
S8	3 785	30,4	3
S9	2 785	33,6	4
S10	5 608	36,0	3
S11	4 753	75,9	7
S12	6 516	77,0	5
S13	7 552	81,3	5
S14	5 166	86,2	7
S15	4 652	87,5	8
S16	4 021	39,2	4

S17.	7 509	42,9	3
S18	5 750	43,6	4
S19	5 750	47,2	4
S20	2 370	47,7	6
S21	3 490	65,7	7
S22	8 759	66,3	4
S23	3 325	69,2	7
S24	7 583	71,1	5
S25	6 247	71,4	5
S26	3 170	62,4	6
S27	8 406	63,4	4
S28	3 915	64,9	7
S29	3 323	65,5	7
S30	4 892	50,2	5
S31	9 692	51,5	3
S32	3 800	52,1	5
S33	4 896	65,1	5
S34	7 612	61,1	4
S35	6 737	99,5	7
S36	4 820	25,4	2
S37	3 480	27,3	3
S38	3 775	30,4	3
S39	2 775	33,6	4
S40	5 600	36	3
S41	4 743	75,9	7
S42	6 507	77	5
S43	7 542	81,3	5
S44	5 146	86,2	7
S45	4 642	87,5	8
S46	4 011	39,2	4
S47	7 500	42,9	3
S48	5 740	43,6	4
S49	5 750	47,2	4

After rejecting the first model tested, the analyst tries a second in which the sales volume of a branch is explained by potential customers that it serves, which he defines as the estimated population of the agglomeration served divided by the number of stores of the same type as the chain studied for this city. It is therefore necessary to define a new variable POPL = POP/ NBM and to postulate model 2: VNT = b0 + b1 POPL. With Excel's Graphical Assistant, draw the point cloud by taking as x-axis (X) the variable "POPL" and as y-axis (Y) the variable "VNT

Does the new point cloud suggest a linear connection between these two variables: VNT and POPL? Explain

Answer 1

According to,

model 2: VNT = b0 + b1 POPL. With Excel's Graphical Assistant, draw the point cloud by taking as x-axis (X) the variable "POPL" and as y-axis (Y) the variable "VNT

I made a graph on excel and it looks like that

POPLVS VNT 12000 10000 8000 minum 6000 4000 2000 о 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 POPL -VNT (thousands of $)

So, its clear that its not a linear connection.

Again to prove that its not a linear connection, I made a second attempt. In this try, with the help of the POPL and VNT data we try to find the B1 and B0 (from: VNT = b0 + b1 POPL)

If the b0 and b1 found to be constant for the all given data then its concluded that VNT and POPL has a linear connection.

Formula for b1 and b0 are as follows..

(61)1 = (VNT)2 – (VNT)1/(POPL)2 – (POPL)1)

(b0)1 = (VNT)1 - {(b1)1*(POPL)1}

Branches	POP (in thousand)	NBM	POPL	VNT (thousands of $)	B1	B0
s1	895	8	112	4197	110	-8073
S2	926	8	116	4622	21	2144
S3	963	8	120	4721	104	-7772
S4	984	7	141	6817	-45	13079
S5	995	7	142	6747	127	-11247
S6	254	2	127	4830	37	103
S7	273	3	91	3490	29	892
S8	304	3	101	3785	58	-2061
S9	336	4	84	2785	78	-3802
S10	360	3	120	5608	74	-3259
S11	759	7	108	4753	39	558
S12	770	5	154	6516	120	-12036
S13	813	5	163	7552	60	-2281
S14	862	7	123	5166	37	569
S15	875	8	109	4652	55	-1415
S16	392	4	98	4021	78	-3575
S17.	429	3	143	7509	52	111
S18	436	4	109	5750	0	5750
S19	472	4	118	5750	88	-4609
S20	477	6	80	2370	78	-3832
S21	657	7	94	3490	73	-3389
S22	663	4	166	8759	81	-4706
S23	692	7	99	3325	98	-6387
S24	711	5	142	7583	-2227	324215
S25	714	5	143	6247	79	-5078
S26	624	6	104	3170	96	-6822
S27	634	4	159	8406	68	-2414
S28	649	7	93	3915	-691	67950
S29	655	7	94	3323	230	-18177
S30	502	5	100	4892	67	-1870
S31	515	3	172	9692	87	-5300
S32	521	5	104	3800	42	-592
S33	651	5	130	4896	120	-10786
S34	611	4	153	7612	82	-4989
S35	995	7	142	6737	127	-11257
S36	254	2	127	4820	37	93
S37	273	3	91	3480	29	882
S38	304	3	101	3775	58	-2071
S39	336	4	84	2775	78	-3817
S40	360	3	120	5600	74	-3287
S41	759	7	108	4743	39	546
S42	770	5	154	6507	120	-12027
S43	813	5	163	7542	61	-2332
S44	862	7	123	5146	37	638
S45	875	8	109	4642	55	-1425
S46	392	4	98	4011	78	-3587
S47	429	3	143	7500	52	98
S48	436	4	109	5740	1	5619
S49	472	4	118	5750	49	0

I found that b0 and b1 from the S1 branch to S49 branch are not constant and varies too much.

Hence proved that two variable VNT and POPL are not in linear conncetion.