Question

all parts!

7. 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. I (a) Test at the 1% significance level whether, on average, EES reduces CO emissions. (11 marks) Before 1.60 0.30 3.80 6.20 3.60 1.50 2.00 2.60 0.15 0.06 0.60 0.03 0.10 0.19 After 0.15 0.20 2.80 3.60 1.00 0.50 1.60 1.60 0.06 0.16 0.35 0.01 0.00 0.00 (b) 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). (4 marks) (c) Interpret the confidence interval obtained in part (b). Does this interval support the conclusion of the hypothesis tost in part (a)? Justify your answer. (4 marks: 2+2)

Answer 1

(a)

The manufacturer of an Engine Energizer System (EES) claims that it proves 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. The test result performed on a random sample of 14 vehicles.

The data of CO (parts per million) levels both before and after installation of EES is given below:

Before	After	Difference
1.60	0.15	1.45
0.30	0.20	0.10
3.80	2.80	1.00
6.20	3.60	2.60
3.60	1.00	2.60
1.50	0.50	1.00
2.00	1.60	0.40
2.60	1.60	1.00
0.15	0.06	0.09
0.06	0.16	-0.10
0.60	0.35	0.25
0.03	0.01	0.02
0.10	0.00	0.10
0.19	0.00	0.19

The mean CO emission level before installation of EES is,

mean 1.60 + ... + 0.19 14 = 1.624

The mean CO emission level after installation of EES is,

mean 0.15 + ... +0.00. 14 = 0.859

Here we want to test whether the average CO emission level get reduced after EES installation.

Hence our null hypothesis of chief interest are:

он
: There is no difference between CO emission levels before and after installation of EES. i.e.
Pr
= 0

$H_{1}$ : There is significant reduction between CO emission levels before and after installation of EES. i.e. > 0

Pr

Test statistic: $t = \frac{\bar{d} - \mu _{d}}{\frac{sd}{\sqrt{n}}}$

Here $\bar{d}$ = The sample mean difference = 1.624 - 0.859 = 0.765

Pr
= Poplulation mean difference = 0 (under
он
)

sd = Standard deviation of the differences = 0.909

$\frac{sd}{\sqrt{n}} = 0.909/\sqrt{14} = 0.243$

Then  $t = \frac{0.765 - 0}{0.243} = 3.148$

The critical value of t is

Here $t_{obs}$ =3.148

So, $t_{obs}$ > 2.65

Ans: Hence, we can conclude that there is significant reduction in CO emission levels after installation of EES.

(b)

The confidence interval around the difference in means

$\bar{d} \pm (t_{\alpha , n-1} * \frac{sd}{\sqrt{n}})$

Here,

$\bar{d}$ = The sample mean difference = 1.624 - 0.859 = 0.765

Pr
= Poplulation mean difference = 0 (under
он
)

sd = Standard deviation of the differences = 0.909

$\frac{sd}{\sqrt{n}} = 0.909/\sqrt{14} = 0.243$

The critical value of t is

Hence, $\bar{d} \pm (t_{\alpha , n-1} * \frac{sd}{\sqrt{n}}) = 1.624 \pm (2.650*0.243)$ = (0.98, 2.268)

The confidence interval around the difference in means is (0.98, 2.268)

(c)

Ans: there is a 99% chance that the confidence interval (0.98, 2.268) we calculated contains the true population mean difference i.e. if we take samples 100 times from this population then 99% of the times the confidence interval (0.98, 2.268) we calculated contains the true population mean difference.

Since the sample mean difference = 0.765 does not lie in the confidence interval of (0.98, 2.268) we reject the null hypothesis that the true population mean difference is 0 and conclude that the true population mean difference is not 0 i.e. there is a significant reduction in CO emission levels after installation of EES.

Ans: The interpretation of confidence interval supports the conclusion of hypothesis test in part (a).