2024 YIJC H1 Q11

Bags of carrots are said to have a mass of \(2\) kg. An inspector claims that the mean mass is not \(2\) kg. To test this claim, a random sample of \(40\) bags is taken and the masses, \(x\) kg, are summarised as follows.

\[\sum x = 86 \qquad \sum x^2 = 199\]

1. Calculate unbiased estimates of the population mean and variance.[2]
2. Test the inspector's claim at the \(10\)% significance level.[4]
3. Explain what you understand by the phrase 'at the \(10\)% significance level' in the context of the question.[1]

Bags of radishes are also said to have a mass of \(2\) kg. The inspector decides to carry out a hypothesis test, at the \(10\)% significance level, on a random sample of \(n\) bags of radishes to find out if the mean mass of bags is more than \(2\) kg. He assumes that the masses of bags of radishes are normally distributed with standard deviation \(0.45\) kg. It is given that the sample mean mass is \(2.1\) kg.

4. Find the set of possible values of \(n\) that would lead to the conclusion that the mean mass of bags of radishes is more than \(2\) kg.[4]