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
Solution
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 ：\(\boxed{\text{(a) The probability that a randomly chosen battery has a life greater than 20.2 hours is approximately 0.8413.}}\)
Step 8 ：\(\boxed{\text{(b) The probability that two randomly chosen batteries have a life of more than 22 hours is approximately 0.0112.}}\)
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Solution
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 ：\(\boxed{\text{(a) The probability that a randomly chosen battery has a life greater than 20.2 hours is approximately 0.8413.}}\)
Step 8 ：\(\boxed{\text{(b) The probability that two randomly chosen batteries have a life of more than 22 hours is approximately 0.0112.}}\)
