2024 NYJC H1 Q11

Observations over a long period of time have shown that the mean daily temperature of a country during the month of June is \(23.9^\circ C\). A meteorologist believes that the daily temperature, \(x^\circ C\), is increasing due to climate change. She collects a random sample of daily temperature readings for \(50\) days in June over a few years and her results are summarised as follows:

\[\sum(x - 20) = 228.25 \quad \quad \sum(x - 20)^2 = 2049.8796\]

1. Calculate unbiased estimates of the population mean and variance for the daily temperature.[2]
2. Test, at the \(5\)% significance level, whether this information supports the meteorologist's claim.[4]

It is believed that a neighbouring country is also experiencing daily temperature changes during the month of June due to climate change. Observations over a long period of time reveal that the daily temperature of this country is normally distributed with mean \(22.5^\circ C\) and standard deviation \(2.3^\circ C\). The daily temperatures are collected for \(20\) randomly selected days and a \(2\)-tailed test is carried out at \(5\)% level of significance.

3. Find the range of values of the sample mean daily temperature that would lead to the conclusion that the daily temperature of this neighbouring country has changed.[5]