N2024 P2 Q8

Lee has a bird feeder on his balcony. Every morning he spends \(20\) minutes, while he eats his breakfast, counting the number of birds that visit the feeder. Over many months he has found that the mean number of birds visiting the feeder in this time interval is \(17.3\). When the bird feeder was damaged in a storm, Lee replaced it with a new bird feeder.

He suspects that the mean number of birds visiting the new bird feeder while he eats his breakfast has reduced. He decides to check this with a hypothesis test at the \(\alpha\%\) level of significance, where \(\alpha\) is an integer. He records the number of birds, \(x\), visiting the feeder in \(20\) minutes each morning for a random sample of \(n\) mornings.

1. Explain whether Lee should carry out a one-tailed test or a two-tailed test.[1]
2. State hypotheses for Lee's test, defining any parameters that you use.[2]

Here is a summary of the data Lee collected.

\(n = 32 \qquad \sum x = 512 \qquad \sum x^2 = 8702\)

Lee carried out his test and concluded that the null hypothesis should be rejected.

3. Calculate unbiased estimates of the population mean and variance of the number of birds, and determine the minimum possible value of the integer \(\alpha\).[3]
4. State the conclusion to Lee's test in the context of the question.[1]
5. Explain whether using the number of birds for each of \(10\) mornings would have been sufficient for Lee to carry out his hypothesis test.[1]