2023 HCI P2 Q11

Researchers at University \(S\) claimed that Fungus A, a strain of fungi that is typically found in soil and plants, was able to break down polypropylene, a plastic that is hard to recycle, that had been pre-treated with either UV light or heat by at least \(27\) grams per \(100\) grams of polypropylene over \(90\) days.

Researchers at University \(M\) believe that the researchers at University \(S\) overstated their claim and decided to perform a hypothesis test on a random sample of \(120\) batches of polypropylene, each of mass \(100\) grams. The researchers recorded the amount \(x\) grams of polypropylene for each batch that Fungus A was able to break down over a \(90\)-day period. The results are summarised as follows:

\(\sum{\left( x-27 \right)=-81}\), \(\sum{{{\left( x-27 \right)}^{2}}=2070.8}\)

1. Calculate the unbiased estimates of the population mean and variance.[3]
2. Test, at \(5\%\) level of significance, the claim that the population mean mass of polypropylene per \(100\) grams that were broken down is at least \(27\) grams. You should state your hypotheses and define the symbols you use.[4]
3. State, in the context of the question, what you understand by the expression ‘at \(5\%\) level of significance’ found in part (b).[1]
4. Explain if it is necessary for University \(M\) to assume a normal distribution for the mass of polypropylene per \(100\) grams that Fungus A was able to break down for this test to be valid.[1]

University \(S\) responded by experimenting on \(50\) batches of polypropylene, each of mass \(100\) grams, and recorded the amount \(y\) grams of polypropylene for each batch that Fungus A was able to break down over a \(90\)-day period. The null and alternative hypotheses remain the same as the ones stated by University \(M\) in part (b).

Based on the research findings from University \(S\), the population standard deviation of the mass of polypropylene per \(100\) grams that Fungus A was able to break down is known to be \(4\) grams.

5. Find the set of values of sample mean mass, \(\bar{y}\) grams, of polypropylene per \(100\) grams from University \(S\) such that the null hypothesis is not rejected in favour of the alternative hypothesis at \(5\%\) level of significance.[3]