Question 3 (3 marks)
Quality checks are regularly conducted in a factory that produces rechargeable lithium-ion batteries. In a particular factory, the life span of a charged battery has a mean of 21 hours and standard deviation of 0.8 hours.
1
(a) A battery is chosen at random.
Find the probability that the battery has a life greater than 20.2 hours.
(b) Two batteries are chosen at random.
Use the section of the table showing the values of $P(Z<z)$ to find the probability that both batteries have a life of more than 22 hours.
$$\text{Unknown environment 'tabular'}$$

Step 1 ：Calculate the z-scores for 20.2 hours and 22 hours using the formula: $z=\frac{x-\text{mean}}{\text{standard deviation}}$

Step 2 ：For 20.2 hours: $z=\frac{20.2-21}{0.8}=-1.00$

Step 3 ：For 22 hours: $z=\frac{22-21}{0.8}=1.25$

Step 4 ：Use the Z-table to find the probabilities:

Step 5 ：For a life greater than 20.2 hours, find the complement probability: $1-P(Z<-1.00)=1-0.1587=0.8413$

Step 6 ：For two batteries with a life of more than 22 hours, find the complement probability and square it: $(1-P(Z<1.25){)}^2=(1-0.8944{)}^2=0.0112$

Step 7 ：$\boxed{{\displaystyle \text{(a) The probability that a randomly chosen battery has a life greater than 20.2 hours is approximately 0.8413.}}}$

Step 8 ：$\boxed{{\displaystyle \text{(b) The probability that two randomly chosen batteries have a life of more than 22 hours is approximately 0.0112.}}}$