Observations over a long period of time have shown that the mean daily temperature of a country duri

Question

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: $$\sum(x - 20) = 228.25 \quad \quad \sum(x - 20)^2 = 2049.8796$$ Calculate unbiased estimates of the population mean and variance for the daily temperature.[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. 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]

Accepted Answer

(a)$\bar{x}=24.565$,$s^2=20.6$ (c)$\bar{x} \leq 21.5 $ or $\bar{x} \geq 23.5$

2024 NYJC H1 Q11

2024 NYJC H1 Q11