A refinery makes two grades of gasoline, regular and premium. The advertised oct

Question

A refinery makes two grades of gasoline, regular and premium. The advertised octane ratings are 87 for regular gasoline and 89 for premium gasoline.

(a) The quality engineer at the refinery asks for 10 samples from one of the two types of gasoline. She does not know for sure whether the samples are from the regular batch or the premium batch.

She devises the hypothesis test H0 : µ 87 H1 : µ > 87 and sets the confidence level to be 0.995. Suppose that the mean of the 10 samples is 88.3 and the standard deviation is 1.0. What is her conclusion for the hypothesis test?

(b) Suppose that a gas station owner has his own octane test kit and rule for accepting a tanker-truck of premium gasoline. The owner knows from past shipments that the distributions of octane ratings are

Xregular N(87, 1)

Xpremium N(89, 1)

Although the owner may not think about it explicitly, his hypothesis test is H0 : µ = 87 H1 : µ = 89 The owner takes one sample from the tanker-truck. If the octane measurement is greater than 88.5, then he will accept the shipment as premium gasoline. What is the probability that the owner accepts a shipment of regular gasoline as premium (i.e. what is )? What is the probability that he declines a shipment of premium gasoline, claiming that he thinks it is regular (i.e. what is )? Use the Normal distribution for this problem.

Explanation / Answer

Solution:

From the given information
= 87, x = 88.3
n = 10, s = 1.0
Test hypothesis:
H0: = 87
H1: > 87
Level of significance :
The given level of significance is 0.995

Test Statistic t = x - /(s/n)
= 88.3- 8.7 / (1.0/sqrt(10))
= 4.1110
df = n-1 = 10-1 = 9
P-value = 0.0013

Conclusion: Since the p value is less than the level of significance 0.9995. Thus, there is enough evidence to reject the null hypothesis at 1% level of significance.
Therefore, conclude that the average samples of gasoline is included for regular gasoline is exceeds the premium gasoline.

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