Reference no: EM132630148

EG-285 Statistical Techniques in Engineering - Swansea University

Background.

Data Set 1.

The following data set gives the times (in years) between failures of air conditioning equipment on a Boeing 720 aircraft.

1.22	0.63	1.07	0.88
1.26	0.72	1.12	0.89
1.28	0.75	1.13	0.95
1.34	0.76	1.19	1.04
1.35	0.86	1.21	1.75

Data Set 2.

Semiconductors (chips) are produced on wafers that contains hundreds of chips. The wafer yield is defined to be the proportion of these chips that are acceptable for use and control engineers aim to maximise this yield during manufacturing. This yield is greatest when the thickness of the coating material to the wafer is uniform as measured using the standard deviation in coating thickness (Y). Control engineers are having difficulties producing a uniform coating at their plant. So an experiment was run to try and gain control over this uniformity.

In this experiment the effect of the 3 process variables (speed X1, pressure X2 and distance X3) on the standard deviation in wafer thickness was studied. Three different values were used for these process variables and were coded -1, 0 and +1 to signify low, medium and high amounts for these variables:

Speed, X1	Pressure, X2	Distance, X3	Standard Deviation in coating thickness, Y
-1	-1	-1	24
0	-1	-1	120.3
1	-1	-1	213.7
-1	0	-1	86
0	0	-1	136.6
1	0	-1	340.7
-1	1	-1	112.3
0	1	-1	256.3
1	1	-1	271.7
-1	-1	0	81
0	-1	0	101.7
1	-1	0	357
-1	0	0	171.3
0	0	0	372
1	0	0	501.7
-1	1	0	264
0	1	0	427
1	1	0	730.7
-1	-1	1	220.7
0	-1	1	239.7
1	-1	1	422
-1	0	1	199
0	0	1	485.3
1	0	1	673.7
-1	1	1	176.7
0	1	1	501
1	1	1	1010

Questions on Data Set 1.

1. To model the times between failures using the exponential distribution, how should the actual times to failure be distributed?
2. Using the exponential distribution, work out the probability that the time between failures will be:
a. Less than 1.07 years.
b. More than 1.34 years.
c. Between 0.5 and 0.75 years.

3. Test, using a probability plot for the exponential and Weibull distributions, whether times between failures are exponentially distributed (as opposed to Weibull distributed).
4. Using the Weibull distribution estimate:
a. The 63rd quantile of the times between failure.
b. The value for β.
5. Using the t test, produce a 95% confidence interval for the mean. Use your answer to explain if this mean is significantly different from 0.9 at the 5% significance level. Explain fully any assumptions behind the t test and the implications such assumptions have for the power of the test.
6. Explain any other distribution your estimated Weibull distribution approximates.

Questions on Data Set 2.

1. Using the above data set and the technique of multiple least squares, estimate the β parameters of the following response surface model:

Y= β₀ + ∑_i=1³ βiXi + ∑_i=1³ βiiX- + ε
 
where Y is the standard deviation in coating thickness, and ε is the prediction error or residual. X1 is speed, X2 is pressure and X3 is distance. How good a fit is this model to the data?

2. Identify which variables are statistically significant at the 10% significance level. State any assumptions that are needed in assessing such statistical significance, and where appropriate construct suitable plots to validate these assumptions.

3. Estimate the parameters of the simplified version of the model shown in question 1 above, that includes only the statistically significant variables (at the 10% significance level). Using these estimates for β, describe in words, the effect of a unit change in distance.

4. Using your simplified model, what test conditions (i.e. what speed, distance and pressure) would you recommend if it is desired to maximise the yield of the process. Make full use of any suitable 2D or 3D scatter plots to explain these conditions.