2025 RVHS P2 Q11

A family business sells strings which are manually cut into stipulated lengths. A client placed an order for \(10\,000\) strings of length \(100\) centimetres. The business owner got the workers to cut out the \(10\,000\) strings. To ensure quality, a random sample of \(40\) strings is chosen from the \(10\,000\) strings for measurement and a constant \(a\) was subtracted from the lengths in centimetres to produce the following summary data.

\[\sum (x-a) = 10.316 \quad \sum (x-a)^2 = 41.7\]

1. Explain in the context of the question what it means for the sample to be random.[1]
2. Find the unbiased estimates of the population mean and variance.[2]
3. Using the data, a test was conducted at \(10\%\) level of significance and found insufficient evidence to conclude that the mean is not \(100\) centimetres. State the hypotheses for the test, defining any parameters that you use. State the critical region and use it to find the range of possible values for the sample mean, giving your answer correct to \(2\) decimal places.[6]
4. Explain if the business owner needs to make any assumption about the distribution of the length of strings for the calculations in part (c) to be valid.[1]
5. The \(p\)-value found from the test in part (c) is \(0.10286\). Write down the \(p\)-value if the test was conducted at \(6\%\) level of significance on whether the strings are shorter than \(100\) centimetres. State the conclusion of this test.[2]