2020 VJC P2 Q9

A company sells a certain type of fitness tracker batteries. The company claims that the mean lifetime of the batteries is at least \(2000\) hours. The production manager of the company wishes to take a random sample of batteries from the thousands produced in a day at his factory, for quality control purposes.

1. State what it means for a sample to be random in this context.[1]

A random sample of \(200\) batteries is taken and it is found that the mean lifetime was \(1995\) hours and the standard deviation was \(25.5\) hours.

2. Test, at the \(1\%\) significance level, whether there is any evidence to doubt the company’s claim.[6]
3. State two assumptions or approximations that are involved when carrying out the significance test using the above sample data.[2]

The company wishes to test whether the average lifetime of a second type of battery is longer than \(2000\) hours. It can be assumed that the lifetime of the second type of battery has a normal distribution with variance \({{\sigma }^{2}}\). \(8\) randomly chosen batteries of the second type \(2\) are tested and it is found that their lifetimes, in hours, are as follows.

\({2001.5, \qquad 1997.3, \qquad 1989.0, \qquad 2018.7, \qquad 2030.7, \qquad 2023.0, \qquad 2027.5, \qquad 2013.3}\)

The company carries out a hypothesis test at the \(5\%\) level of significance.

4. Explain, with justification, how the population variance of the lifetimes, \({{\sigma }^{2}}\), will affect the conclusion.[4]