Solution — 2020 Question 12 Sampling| Hypothesis Testing

Let $X$ denote the random variable of the height (in cm) of a randomly chosen plant
Let $\mu$ denote the population mean height (in cm) of a randomly chosen plant

${\mathrm{H}}_0:\mu =28.5$
${\mathrm{H}}_1:\mu \ne 28.5$

From the sample, $\bar{x}=27.7,{\sigma }^2=6.6,n=40$

Under $H_0,$ $$\bar{X}\sim \mathrm{N}(28.5,\frac{{6.6}^2}{40})$$ $$Z=\frac{\bar{X}-28.5}{\sqrt{\frac{{6.6}^2}{40}}}\sim \mathrm{N}(\mathrm{0,1})$$ approximately by the Central Limit Theorem since $n=40$ is large

From GC,
$p$-value = $0.048900\le 0.05\Rightarrow \text{Reject }{\mathrm{H}}_0$

Hence there is sufficient evidence at the $5\%$ level of significance to conclude that the mean height of plants in region A differs from 28.5 cm $∎$

${\mathrm{H}}_0:\mu =28.5$
${\mathrm{H}}_1:\mu \ne 28.5$

Under $H_0,$ $$\bar{X}\sim \mathrm{N}(28.5,\frac{{13.476}^2}{60})$$ $$Z=\frac{\bar{X}-28.5}{\sqrt{\frac{{13.476}^2}{60}}}\sim \mathrm{N}(\mathrm{0,1})$$ approximately by the Central Limit Theorem since $n=60$ is large

From GC, $p\text{-value}=0.034855$

Since there is insufficient evidence to reject ${\mathrm{H}}_0$: