You are probably all familiar with the Challenger Space Shuttle accident of January 20, 1986. I will leave out all the interesting substantive details about the accident and focus instead on the data about O-rings that was available at the time of the accident.
Well, I do need a little background, so here goes. The space shuttle uses two booster rockets and each rocket is made of several pieces whose joints are sealed by rubber O-rings. The O-rings are supposed to stop the hot gases that are inside the booster from escaping. Each booster contains 3 primary O-rings. There was data about the O-rings on 23 previous flights of space shuttles. Here is that data.
Temp Fail Temp Fail 53 2 70 1 57 1 70 1 58 1 72 0 63 1 73 0 66 0 75 0 67 0 75 2 67 0 76 0 67 0 76 0 68 0 78 0 69 0 79 0 70 0 81 0 70 0
On the day of the fated flight, the forecast launch temperature was
F. The O-rings were known to be sensitive to temperature.
The previous coldest launch temperature was 
. There was
discussion about the cold temperature before launch, and the engineers
did look at the data, but only the data for the failures.
They produced a plot that looked something like this.
402 > chal <- read.table("chal.dat", head=T)
402 > attach(chal, pos=1)
402 > ls()
[1] ".Last.value" "Fail" "Temp"
402 > n <- rep(6, length(Fail))
402 > plot(Temp[Fail >0], Fail[Fail>0])
Based on the erroneous assumption that the flights with no failures did not add any information, they noted that there did not appear to be any relationship between failures and temperature.
Had they produced a plot of the full data, which follows, they might have behaved differently.
402 > plot(Temp, Fail)
In addition before each flight the O-rings were given a pressure test. The pressure at which that test was conducted increased over time, and it is possible that the higher pressure test could have caused more failures. That data, and a lot more discussion, is available in the paper Risk Analysis of the Space Shuttle: Pre- Challenger Prediction of Failure, by S.R Dalal, E.B. Fowlkes, and B. Hoadley, in V. 84, pp.945--957, of the Journal of the American Statistical Association (JASA). You should be able to read most of that article. It is well written and easy to follow.