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.

KS Comments:

Students need to be careful to leave the answers in set notation since the question requests for a set of values. (iii) requires them to answers in the question’s context.

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