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

H_0: \mu = 14

H_1: \mu \ne 14


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 \}

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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