Question

(Please solve manually.)  Every April Americans and Canadians fill out their tax return forms. Many turn to tax preparation companies to do this tedious job. The question arises, Are there differences between companies? In an experiment two of the largest companies were asked to prepare the tax returns of a sample of 55 taxpayers. The amounts of tax payable were recorded. Can we conclude that company 1’s service results in higher tax payable?

Company 1	Company 2
12859	9449
6082	5694
8604	7457
9230	7725
10405	10163
12580	13954
5882	5846
7555	5666
8492	9293
10087	9167
7291	7430
9907	10178
10352	11333
8746	7170
6568	7943
10648	14047
4990	1069
10037	9873
10187	8185
11852	9683
10154	11126
9287	10291
8561	8979
10895	9468
6821	8422
8444	7213
9590	8596
10501	9544
8625	10534
13729	12985
11784	10150
5655	4675
10696	8594
11538	15400
8183	7262
8647	8008
6547	4402
8067	6818
4051	3878
10201	7279
7145	6374
13944	15481
10870	7491
9901	4695
5334	8507
9516	9712
7782	5108
6446	5291
7519	7249
10070	6548
11815	9631
6069	7798
12073	13252
6466	7274
9782	11055

Answer 1

company 1	company 2
12859	9449
6082	5694
8604	7457
9230	7725
10405	10163
12580	13954
5882	5846
7555	5666
8492	9293
10087	9167
7291	7430
9907	10178
10352	11333
8746	7170
6568	7943
10648	14047
4990	1069
10037	9873
10187	8185
11852	9683
10154	11126
9287	10291
8561	8979
10895	9468
6821	8422
8444	7213
9590	8596
10501	9544
8625	10534
13729	12985
11784	10150
5655	4675
10696	8594
11538	15400
8183	7262
8647	8008
6547	4402
8067	6818
4051	3878
10201	7279
7145	6374
13944	15481
10870	7491
9901	4695
5334	8507
9516	9712
7782	5108
6446	5291
7519	7249
10070	6548
11815	9631
6069	7798
12073	13252
6466	7274
9782	11055
9073.854545(mean)	8553(mean)
2276.23729(standard deviation)	2853.51321(standard deviation)

the ratio between the two standard deviation is 0.797696 which is not more than 1.4 thus it can be assumed that the two population has the same standard deviation.

The test to compare the tax payment is as follows.

Thus we cannot conclude that company 1’s service results in higher tax payable.

[NOTE: The ratio of the two standard deviation is important to know because if its value is less than 1.4 only the we can use this method ]

[In case you need the results to calculate the standard deviations manually the values are given bellow.

• x1i=499062 x1bar= 499062/55=9073.854545 x1bar2=82334835.49
• x2i=470415 x2bar=470415/55=8553  x1bar2=73153809

x12	x22
165353881	89283601
36990724	32421636
74028816	55606849
85192900	59675625
108264025	103286569
158256400	194714116
34597924	34175716
57078025	32103556
72114064	86359849
101747569	84033889
53158681	55204900
98148649	103591684
107163904	128436889
76492516	51408900
43138624	63091249
113379904	197318209
24900100	1142761
100741369	97476129
103774969	66994225
140469904	93760489
103103716	123787876
86248369	105904681
73290721	80622441
118701025	89643024
46526041	70930084
71301136	52027369
91968100	73891216
110271001	91087936
74390625	110965156
188485441	168610225
138862656	103022500
31979025	21855625
114404416	73856836
133125444	237160000
66961489	52736644
74770609	64128064
42863209	19377604
65076489	46485124
16410601	15038884
104060401	52983841
51051025	40627876
194435136	239661361
118156900	56115081
98029801	22043025
28451556	72369049
90554256	94322944
60559524	26091664
41550916	27994681
56535361	52548001
101404900	42876304
139594225	92756161
36832761	60808804
145757329	175615504
41809156	52911076
95687524	122213025

x12 =4808203832 and (x12)/55=87421887.9

x22 =4463156527 and (x22)/55=81148300.49 ]