2016 VCE Maths Methods Mini Test 6

Question 1 [2016 Exam 2 Section B Q3]

A school has a class set of 22 new laptops kept in a recharging trolley. Provided each laptop is correctly plugged into the trolley after use, its battery recharges.
On a particular day, a class of 22 students uses the laptops. All laptop batteries are fully charged at the start of the lesson. Each student uses and returns exactly one laptop. The probability that a student does not correctly plug their laptop into the trolley at the end of the lesson is 10%. The correctness of any student's plugging-in is independent of any other student's correctness.

a. Determine the probability that at least one of the laptops is not correctly plugged into the trolley at the end of the lesson. Give your answer correct to four decimal places. 2 marks

b. A teacher observes that at least one of the returned laptops is not correctly plugged into the trolley.
Given this, find the probability that fewer than five laptops are not correctly plugged in. Give your answer correct to four decimal places. 2 marks

c. The time for which a laptop will work without recharging (the battery life) is normally distributed, with a mean of three hours and 10 minutes and standard deviation of six minutes. Suppose that the laptops remain out of the recharging trolley for three hours.
For any one laptop, find the probability that it will stop working by the end of these three hours. Give your answer correct to four decimal places. 2 marks

d. A supplier of laptops decides to take a sample of 100 new laptops from a number of different schools. For samples of size 100 from the population of laptops with a mean battery life of three hours and 10 minutes and standard deviation of six minutes, \(\hat{P}\) is the random variable of the distribution of sample proportions of laptops with a battery life of less than three hours.
Find the probability that \(\Pr(\hat{P} \ge 0.06 | \hat{P} \ge 0.05)\). Give your answer correct to three decimal places. Do not use a normal approximation. 3 marks

e. It is known that when laptops have been used regularly in a school for six months, their battery life is still normally distributed but the mean battery life drops to three hours. It is also known that only 12% of such laptops work for more than three hours and 10 minutes.
Find the standard deviation for the normal distribution that applies to the battery life of laptops that have been used regularly in a school for six months, correct to four decimal places. 2 marks

f. The laptop supplier collects a sample of 100 laptops that have been used for six months from a number of different schools and tests their battery life. The laptop supplier wishes to estimate the proportion of such laptops with a battery life of less than three hours.
Suppose the supplier tests the battery life of the laptops one at a time.
Find the probability that the first laptop found to have a battery life of less than three hours is the third one. 1 mark

g. The laptop supplier finds that, in a particular sample of 100 laptops, six of them have a battery life of less than three hours.
Determine the 95% confidence interval for the supplier's estimate of the proportion of interest. Give values correct to two decimal places. 1 mark

h. The supplier also provides laptops to businesses. The probability density function for battery life, \(x\) (in minutes), of a laptop after six months of use in a business is \(f(x) = \begin{cases} \frac{(210-x)e^{\frac{x-210}{20}}}{400} & 0 \le x \le 210 \\ 0 & \text{elsewhere} \end{cases}\)

i. Find the mean battery life, in minutes, of a laptop with six months of business use, correct to two decimal places. 1 mark

ii. Find the median battery life, in minutes, of a laptop with six months of business use, correct to two decimal places. 2 marks