- The lifetime, \(L\) hours, of a battery has a normal distribution with mean 18 hours and standard deviation 4 hours.
Alice's calculator requires 4 batteries and will stop working when any one battery reaches the end of its lifetime.
- Find the probability that a randomly selected battery will last for longer than 16 hours.
At the start of her exams Alice put 4 new batteries in her calculator. She has used her calculator for 16 hours, but has another 4 hours of exams to sit.
- Find the probability that her calculator will not stop working for Alice's remaining exams.
Alice only has 2 new batteries so, after the first 16 hours of her exams, although her calculator is still working, she randomly selects 2 of the batteries from her calculator and replaces these with the 2 new batteries.
- Show that the probability that her calculator will not stop working for the remainder of her exams is 0.199 to 3 significant figures.
After her exams, Alice believed that the lifetime of the batteries was more than 18 hours. She took a random sample of 20 of these batteries and found that their mean lifetime was 19.2 hours.
- Stating your hypotheses clearly and using a \(5 \%\) level of significance, test Alice's belief.