All solutions here are SUGGESTED. Mr. Teng will hold no liability for any errors. Comments are entirely personal opinions.

(i)

Let $\mu$ be the population mean tail length of the squirrels, in cm

$H_0: \mu = 14$

$H_1: \mu \ne 14$

(ii)

Under $H_0, ~\bar{X} ~\sim~ \mathrm{N}(14, \frac{3.8^2}{20})$

Test Statistic, $z = \frac{\bar{X} - \mu_0}{s/ \sqrt{n}} ~\sim~ \mathrm{N}(0,1)$

For $H_0$ to be not rejected at 5% level of significance,

$|z| = |\frac{\bar{x} - 14}{3.8/ \sqrt{20}}| \textless 1.95996$

$-1.95996 (\frac{3.8}{\sqrt{20}}) + 14 \textless \bar{x} \textless 1.95996 (\frac{3.8}{\sqrt{20}}) + 14$

$12.3 \textless \bar{x} \textless 15.7$

Thus the set of values is $\{ \bar{x} \in \mathbb{R}| 12.3 \textless \bar{x} \textless 15.7 \}$

(iii)
If $\bar{x} = 15.8, ~H_0$, will be rejected from results in (ii). There is sufficient evidence at 5% level of significance to conclude that the squirrels on the island do not have the same tail length as the species known.