Solution — 2025 Question 12 Sampling| Hypothesis Testing

Let $X$ denote the random variable of the salinity (in g/kg) of a randomly chosen seawater specimen
Let $\mu$ denote the population mean salinity (in g/kg) of a randomly chosen seawater specimen

${\mathrm{H}}_0:\mu =35$
${\mathrm{H}}_1:\mu >35$

Under $H_0,$ $$\bar{X}\sim \mathrm{N}(35,\frac{{\frac{319}{395}}^2}{80})$$ $$Z=\frac{\bar{X}-35}{\sqrt{\frac{{\frac{319}{395}}^2}{80}}}\sim \mathrm{N}(\mathrm{0,1})$$ approximately by the Central Limit Theorem since $n=80$ is large

From GC,
$p$-value = $0.023264\le 0.05\Rightarrow \text{Reject }{\mathrm{H}}_0$

Hence there is sufficient evidence at the $5\%$ level of significance to conclude that the mean salinity of seawater in the area has increased $∎$

${\mathrm{H}}_0:\mu =35$
${\mathrm{H}}_1:\mu \ne 35$

where $\mu$ g/kg is the population mean salinity in the different area.

Since there is insufficient evidence to reject ${\mathrm{H}}_0$ at the $5\%$ significance level,

Since $\bar{x}-\mu =35.08-35>0,$

Let ${\sigma }_x$ be the standard deviation of the sample