Question

1. The manufacturer of an Engine Energizer System (EES) claims that it improves gas mileage and reduces emissions in automobiles by using magnetic-free energy to increase the amount of oxygen in the fuel for greater combustion efficiency. Following are test results, performed under international and U.S. government agency standards, on a random sample of 14 vehicles. The data (also see file M09_Gas_Paired.txt) give the carbon monoxide (CO) levels, in parts per million, of each vehicle tested, both before installation of EES and after installation.

Before After 1.60 0.15 0.30 0.20 3.80 2.80 6.20 3.60 3.60 1.00 1.50 0.50 2.00 1.60 2.60 1.60 0.15 0.06 0.06 0.16 0.60 0.35 0.03 0.01 0.10 0.00 0.19 0.00

Test at the 1% significance level whether, on average, EES reduces CO emissions. Obtain a 99% confidence interval for the difference between the mean CO emissions before and after installation of EES corresponding to the test in part (a). Interpret the confidence interval obtained in part (b). Does this interval support the conclusion of the hypothesis test in part (a)? Justify your answer. (4 marks: 2+2)

Answer 1

Answer a)

Paired Sample t-Test has been used to test the claim. Foll

Thus, at 0.01 significance level, there is enough evidence to claim that on average, EES reduces CO emissions.

Answer b)

Step 1: Find α/2
Level of Confidence = 99%
α = 100% - (Level of Confidence) = 1%
α/2 = 0.5% = 0.005


Step 2: Find tα/2
Calculate tα/2 by using t-distribution with degrees of freedom (DF) as n - 1 = 14 - 1 = 13 and α/2 = 0.005 as right-tailed area and left-tailed area.

tα/2 = 3.01207

Step 3: Calculate Confidence Interval
Confidence Formula: [d̄ - tα/2•(sd/√n) , d̄ + tα/2•(sd/√n)]
Lower Bound = d̄ - tα/2•(sd/√n) = 0.764 - (3.01207)(0.909/√14) = 0.032
Upper Bound = d̄ + tα/2•(sd/√n) = 0.764 + (3.01207)(0.909/√14) = 1.496
Confidence Interval = (0.032, 1.496)

Answer c)

We are 99% confident that the true mean difference between CO emissions before and after installation of EES lies between 0.032 and 1.496

The confidence interval does not contain zero, so null hypothesis is rejected. Thus, this interval support the conclusion of the hypothesis test in part (a).