2021 ACJC P2 Q9

The Particle Filtration Efficiency (PFE) of a mask is a measure of how well a mask filters airborne particles such as pollen or dust. A mask with higher PFE is deemed to be of better efficiency as it filters more particles. A mask with a PFE of \(95\%\) would have met the requirement for surgical masks.

A factory manufactures Brand BEY surgical masks that is known to have expected PFE of \({95.8\%}\). During a routine check of the manufacturing process, the quality control manager suspects that the efficiency of the Brand BEY surgical masks produced is compromised such that the mean PFE is reduced. The PFE, \({x\%}\), of a random sample of \({50}\) masks is taken and the summarised results are as follows.

\({\sum(x - 90) = 289} \qquad \sum(x - 90)^2 = 1670.56\)

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

The manager carries out a hypothesis test at \(1\%\) level of significance.

2. Explain whether there is a need for the manager to make any assumption about the population distribution of the PFE of the masks.[1]
3. State the appropriate hypotheses, defining any symbols that you use. Test whether the manager’s suspicion is justified at \(1\%\) level of significance.[5]

The manager discovered an error in the data collection process of the sample and discarded \(10\) out of the \(50\) readings. From the remaining random sample of \(40\) masks, the manager found that the mean PFE is \(95.5\%\) and the standard deviation is \(k\%\). Given that the manager concludes that there is insufficient evidence to justify his suspicion at \(1\%\) level of significance, find the range of possible values of \(k\) used in calculating the test statistic.[3]