Question

1. Please solve the following problem. The mean emission of all engines of a new design needs to be below 40 ppm (parts per million) if the design is to meet new emission requirements. Ten engines are manufactured for testing purposes, and the emission level of each is determined. The emission data is: 32.4, 33.1, 45.2, 43, 33.2, 38.6, 33.5, 35.7, 25.5, 27.6. Does the data supply sufficient evidence to conclude that this type of engine meets the new standard? Assume we are willing to risk a Type I error with probability=0.01

Answer 1

Given sample size is not large therefore we will use t test for single mean here

The new standard of mean emission for all engine is 40

Hence if mean of emission is greater than 40 the engines will not meat the new standard.

Now

Null hypothesis,

Ho: 40

Alternative hypothesis, $H_{1}:\mu> 40$

Test Statistic under $H_{0}$ is given by

$t=\frac{\bar X -\mu}{s/\sqrt{n}}$

Given of a neur 40 ppm 33.1 45.2 : 43 33.2 38.6 3 3.5 + / / 25.5 The mean emission all engines of design needs to below Emmision data for n=10 engines 32.4 35.7 27.6 probability of Type I Evror, d =0.0| To meet the new standard population mean emmision must not be greater thorn 4 oppm. Null Hypothesis, Hoi 4 = 40 Alternative Hypothesis, H, 4 >40 (Right tailed test) Now, Critical value Tabulated value of t for 10-1=g af at oiol level of significante for single talled test is 2.821

test statistic . The test statßer under no tcal X-M ntni sin Men Calculation deurator and sample standard S. No, 3. 4 Xi 32.4 33:1 45.2 43 33.2 38.6 33.5 35.7 25.5 Sample mem X= X; -X -2.38 --168 10.42 8.22 -1.58 3.82 -1-28 0.92 -9.28 7.18 n (xi-x) 5: 6644 9.8224 108.5764 67.5684 2-4964 14. 5924 1.6384 0.3464 86.1184 51.5524! 341.871 = 347.3 rt X = 34.78 9. llo 27.6 Sample standard deviation 8= I Cxi-x)² Total 347.8 0 341.876 = s = 6.163 Now. こ t 34.78 -40 -2.678 6.163/Ja It = -2.678] Decision Since t =-2-678 < 2.821 it is then Concluded that the null Hypothesis & not rejected It means the sample data does not supply sufficient evidence te conclude that this type of engine do not meet this new standard,
we fail to reject null hypothesis it means mean emission of this type of engine is not greater than 40 .

Therefore this type of engines meet the new standard.