Question

Q1: A transport company wants to compare the fuel efficiencies of the two types of lorry it operates. It obtains data from samples of the two types of lorry, with the following results:

Type	Average mpg	Std devn	Sample size
A	30.1	7.6	13
B	33.2	5.8	17

At 90% confidence level, test the hypothesis that lorries of type B are more efficient than type A. Use critical value approach and assume unequal variation in mpg of two types.

Answer 1

Solution

Let X = mpg of lorry of Type A and Y = mpg of lorry of Type B

Then, X ~ N(µ₁, σ₁²) and Y ~ N(µ₂, σ₂²), where σ₁² and σ₂² are unequal [given, ‘assume unequal variation in mpg of two types’ ] and unknown.

Claim:

Lorries of Type B are more efficient than lorries of Type A

Hypotheses:

Null: H₀: µ1 = µ2 Vs Alternative: H_A: µ1 < µ2

Test Statistic:

t = (Xbar - Ybar)/√[{(s₁²/n₁) + (s₂²/n₂)}] where

Xbar and Ybar are sample averages and s₁,s₂ are sample standard deviations based on n1 observations on X and n2 observations on Y respectively.

Calculations

Summary of Excel calculations is given below:

n1 =	13
n2 =	17
Xbar =	30.1
Ybar =	33.2
s1 =	7.6
s2 =	5.8
s1^2/n1	4.443077
s2^2/n2	1.978824
Sum S	6.4219
sqrt(S)	2.534147
Xbar - Ybar	-3.1
tcal	-1.22329
α	0.1
DF - ν	22
tcrit	-1.32319
ν-calculation
S1 = s1^2/n1	4.443077
S2 = s2^2/n2	1.978824
S = S1 + S2	6.4219
ν1	12
ν2	16
F1	2028
F2	4624
D1= s1^4/F1	1.645078
D2= s2^4/F2	0.244734
D = D1 + D2	1.889812
ν = S^2/D	21.82271
[ν]	22

Distribution, Significance Level , Critical Value

Under H₀, t ~ t_ν, where

ν = {(s₁²/n₁) + (s₂²/n₂)}/{(s₁⁴/n₁²x ν₁) + (s₂⁴/n₂²x ν₂)}; ν₁ = n₁ - 1 and ν₂ = n₂ - 1

Hence, for level of significance α%, Critical Value = lower α% point of t_ν

Using Excel Functions, Statistical TINV, the above are found to be as shown in the above table: [given 90% confidence level, α = 10%, i.e., 0.1]

Decision:

Since tcal > tcrit, H₀ is accepted.

Conclusion:

There is NOT sufficient evidence to suggest that the claim is valid. i.e.,

Lorries of Type B are NOT more efficient than lorries of Type A. ANSWER

DONE