2024 ASRJC H1 Q10

A company claims that the mean weight of a certain type of packaged product is 500 grams. A quality control manager suspects that the company underestimates the mean weight of the packages. To test his hypothesis, the manager takes a random sample of \(40\) packages from the production line. The weights of the packages, \(x\) grams, are summarized as follows:\[\sum(x-500) = 211 \text{ and}\hspace{0.5em} \sum(x-500)^2 = 9479\]

1. Calculate unbiased estimates of the population mean and variance for the weights of the packages.[2]
2. At the \(1\)% significance level, test whether the manager's suspicion is supported by the data.[4]
3. Explain \(p\)-value in the context of the question.[1]

A new random sample of \(80\) packages is taken from a different batch. The mean weight in this new sample is \(k\) grams, and the population variance is known to be \(225\) grams\(^2\). A hypothesis test at the \(2.5\)% significance level gives sufficient evidence to conclude that the mean weight of the packages differs from \(500\) grams.

4. Find the range of possible values of \(k\).[3]