Product and Process Comparisons
7.3. Comparisons based on data from two processes
|Comparing two exponential distributions is to compare the means or hazard rates||
The comparison of two (or more) life distributions is a common
objective when performing statistical analyses of lifetime data.
Here we look at the one-parameter exponential distribution case.
In this case, comparing two exponential distributions is equivalent to comparing their means (or the reciprocal of their means, known as their hazard rates).
|Type II Censored data|
|Definition of Type II censored data||Definition: Type II censored data occur when a life test is terminated exactly when a pre-specified number of failures have occurred. The remaining units have not yet failed. If n units were on test, and the pre-specified number of failures is r (where r is less than or equal to n), then the test ends at tr = the time of the r-th failure.|
|Two exponential samples oredered by time||
Suppose we have Type II censored data from two exponential
distributions with means
We have two samples from these distributions, of sizes
n1 on test with r1 failures and
n2 on test with r2 failures,
respectively. The observations are time to failure and are therefore
ordered by time.
|Test of equality of 1 and 2 and confidence interval for 1/ 2||
A numerical application will illustrate the concepts outlined
For this example,
Ha: 1/ 2 1
Then T1 = 4338 and T2 = 3836.
The estimator for 1 is 4338 / 7 = 619.71 and the estimator for 2 is 3836 / 5 = 767.20.
The ratio of the estimators = U = 619.71 / 767.20 = .808.
If the means are the same, the ratio of the estimators, U, follows an F distribution with 2r1, 2r2 degrees of freedom. The P(F < .808) = .348. The associated p-value is 2(.348) = .696. Based on this p-value, we find no evidence to reject the null hypothesis (that the true but unknown ratio = 1). Note that this is a two-sided test, and we would reject the null hyposthesis if the p-value is either too small (i.e., less or equal to .025) or too large (i.e., greater than or equal to .975) for a 95% significance level test.
We can also put a 95% confidence interval around the ratio of the two means. Since the .025 and .975 quantiles of F(14,10) are 0.3178 and 3.5504, respectively, we have
and (.228, 2.542) is a 95% confidence interval for the ratio of the unknown means. The value of 1 is within this range, which is another way of showing that we cannot reject the null hypothesis at the 95% significance level.