Suppose there is a task at hand to check whether one can really save a bunch...

Suppose there is a task at hand to check whether one can really save a bunch of money on the car insurance by switching from Allstate to GEICO. Their monthly insurance premiums for various car brands are shown in the table below. Conduct a two-sample test of hypothesis at α = 0.05 significance level to determine whether it is worth switching to GEICO.

Car Brands	GEICO	Allstate
Honda	120	136
Toyota	125	135
Nissan	122	135
Ford	107	102
Chevrolet	126	128
Chrysler	110	104
BMW	154	176
Mercedes	160	182
Volkswagen	148	164

Expert Solution

We have the following information

Average Premium of GEICO = (1,172 ÷ 9) = 130.2

Average Premium of Allstate = (1,262 ÷ 9) = 140.2

Car Brands	GEICO	Allstate	Deviation from Mean (GEICO)	Square of Deviation from Mean (GEICO)	Deviation from Mean (Allstate)	Square of Deviation from Mean (Allstate)
Honda	120	136	-10.2	104.5	-4.2	17.8
Toyota	125	135	-5.2	27.3	-5.2	27.3
Nissan	122	135	-8.2	67.6	-5.2	27.3
Ford	107	102	-23.2	539.3	-38.2	1460.9
Chevrolet	126	128	-4.2	17.8	-12.2	149.4
Chrysler	110	104	-20.2	408.9	-36.2	1312.0
BMW	154	176	23.8	565.4	35.8	1280.0
Mercedes	160	182	29.8	886.7	41.8	1745.4
Volkswagen	148	164	17.8	316.0	23.8	565.4
Total	1172	1262		2933.6		6585.6

Standard Deviation of GEICO = √(Square of Deviation from Mean)÷(N – 1); where N is the number of observations

Standard Deviation of GEICO = √(2933.6 ÷ 8) = 19.1

Standard Deviation of Allstat = √(Square of Deviation from Mean)÷(N – 1); where N is the number of observations

Standard Deviation of Allstate = √(6585.6 ÷ 8) = 28.7

Null Hypothesis: There is no significant difference between the average premium of GEICO and Allstate

T-test for mean comparison with unequal standard deviations

T-test statistic = (Mean of GEICO – Mean of Allstate) ÷ √((Standard Deviation of GEICO)²/N1) + ((Standard Deviation of Allstat)²/N2)

N1 and N2 are number of observations. We will be using 95% confidence level

T Stat	-0.71466
P(T<=t) one-tail	0.244245
t Critical one-tail	1.782288
P(T<=t) two-tail	0.488489
t Critical two-tail	2.178813

Since, the calculated T-state value (-0.71466) is less than the critical value (2.178813), so we accept the null hypothesis that there is no significant difference between the average premium of GEICO and Allstate. So, one can say that it is not worth switching to GEICO.

Rahul Sunny answered 2 years ago

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