Question

8) Accelerated lifetime tests are carried out for a batch of 200 thermocouples. The mean lifetime is found to be 936 h with a standard deviation of 28 h. The smallest and largest measurements in the sample are 860 and 1012 h. The measurements are divided into 9 data bins with boundaries at 859.5, 876.5, 893.5, 910.5, 927.5, 944.5, 961.5, 978.5, 995.5, 1012.5. The measurement count in bin 1 from 869.5 days to 886.5 days is 3 and the count in the other successive bins is 8, 27, 39, 46 40, 24, 11 and 2. Apply the chi-squared test to see whether the measurements fit a Gaussian distribution to a a) 99% confidence level, b) a 97.5% confidence level, and c) a 95% confidence level.

Answer 1

Let X denotes Lifetime for a batch of thermocouples

N = 200

E(X) = 936 and S.D(X) = 28

By central lilmit theorem

From the information

Class Interval	Observed Frequency	Probability	Expected Frequecy = N * Probability	Approximate
859.5-876.5	3	P( 859.5 < X < 876.5) = 0.01364	2.7294	3
876.5-893.5	8	P( 876.5 < X < 893.5) = 0.04773	9.5464	10
893.5-910.5	27	P(893.5 < X < 910.5) =0.11669	23.3396	23
910.5-927.5	39	P( 910.5 < X < 927.5) = 0.19950	39.9008	40
927.5-944.5	46	P(927.5 < X < 944.5) = 0.23854	47.7091	48
944.5-961.5	40	P( 944.5 < X < 961.5) = 0.19950	39.9008	40
961.5-978.5	24	P(961.5 < X < 978.5) = 0.11669	23.3396	23
978.5-995.5	11	P(978.5 < X < 999.5) = 0.04773	9.5464	10
995.5-1012.5	2	P( 995.5 < X < 1012.5) = 0.01364	2.7294	3

We have whether the measurement fit a Gaussian distribution(Normal).

we use Chi-square goodness of fit test.

The value of test statistic is

Where Oi : Observed frequency , Ei - Expected Frequency and n = number of classes = 9

Observed Frequency	Expected Frequency	(Oi-Ei)^2 /Ei
3	3	0
8	10	0.4
27	23	0.69565217
39	40	0.025
46	48	0.08333333
40	40	0
24	23	0.04347826
11	10	0.1
2	3	0.33333333
200	200	1.6807971

The value of Chi-square statistic

Critical Values for 8 degrees of freedom for different level of significance

Confidence Interval	Level of significance	Critical Value
99%	0.01	20.0902
97.50%	0.025	17.5345
95%	0.05	15.5073

From the above table we see that all critical values is greater than value of test statistics. we accept Ho at 1%, 2.5% and 5% level of significance.

Conclusion : Measurement fit a Gaussian distribution.