2023 NYJC P2 Q9

The manufacturer of a particular toy drone claims that its average flying time (AFT) is \(12\) minutes. The quality control manager suspects that the AFT is different from what the manufacturer has claimed and decides to carry out a hypothesis test on a sample of these drones.

The flight durations, \(x\) minutes, of a random sample of \(100\) drones are summarised as follows:

\(\sum{x}=1171\) \(\sum{{{x}^{2}}}=14023\)

1. State what it means for a sample to be random in this context.[1]
2. Calculate unbiased estimates of the population mean and variance of the AFT of the toy drones.[2]
3. Test, at the \(5\%\) level of significance, the manufacturer’s claim that the AFT of toy drones is \(12\) minutes. You should state your hypotheses and define any symbols you use.[5]

Over a long period of time, it is established that the AFT of the toy drone is normally distributed with mean and standard deviation of \(12\) minutes and \(2\) minutes respectively. However, changes in the manufacturing process due to an upgrade of the drone’s rechargeable battery, is expected to improve the AFT of the drone. To verify this, a random sample of \(n\) drones is taken with sample mean AFT of \(m\). If \(m\) is greater than some critical value, it will be accepted that the AFT of the toy drone has in fact increased.

4. Find in terms of \(n\), the critical value for a \(5\%\) significance level.[3]