A company is going to launch an exclusive high-end smartphone.
They rely on a bold marketing strategy that offers a lifetime
warranty for replacing the first failure caused by normal use in
each smartphone sold.
The engineering team does a precise accelerated aging test on
150 smartphones and detects that 5 fail. The test is precise in
simulating the real conditions of a lifetime of use.
A 0.0333 probability of failure is just within the long-run
budget for the marketing campaign and everyone in the company is in
good spirits, until someone in the analytics department warns them
that the probability of failure for each smartphone could be much
higher. For example, if the true probability of failure was 0.0533
the budget would be greatly exceeded (by 60% !) and the company
would have to close. What is the probability of having observed the
results of the aging test, or even better results (less failures),
if the true probability of failure is 0.0533
Please use the normal approximation and introduce the answer to
3 decimal places.