The accompanying data are the times to failure (in millions per cycle) of high-speed turbine engine bearings made out of two dif
ferent compounds. These were taken from "Analysis of Single Classification Experiments Based on Censored Samples from the Two-parameter Weibull Distribution" by J.I. McCool (The Journal of Statistical Planning and Inference, 1979) Compound 1 3.03 5.53 5.60 9.30 9.92 12.51 12.95 15.21 16.04 16.84 Compound 2 3.19 4.26 4.47 4.53 4.67 4.69 5.78 6.79 9.37 12.75 (a) Find the 0.84 quantile of the Compound 1 failure times (b) Give the coordinates of the two lower-left points that would appear on a normal plot of the compound 1 data (c) Make back-to-back stem-and-leaf plots for comparing the life length properties of bearings made from Compounds 1 and 2 (d) Make (to scale) side-by-side boxplots for comparing the life lengths for the two compounds. Mark numbers on the plots indicating the locations of their main features (e) Compute the sample means and standard deviations of the two sets of lifetimes (f) Describe what your answers to parts (c), (d), and (e) above indicate about the life lengths of these turbine bearings.
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Notice that
2010 ≡ 1 mod 2009
2011 ≡ 2 mod 2009
2012 ≡ 3 mod 2009
...
4017 ≡ 2008 mod 2009
4018 ≡ 0 mod 2009
So really, <em>S</em> is just the sum of the first 2008 positive integers:
where we invoke the formula
and so <em>S</em> ≡ 0 mod 2009.