2012 H2 Mathematics Paper 2 Question 6

Hypothesis Testing

Answers

{\textrm{H}_0: \mu = 14.0}

{\textrm{H}_1: \mu < 14.0}

{\{ \overline{x} \in \mathbb{R}: 12.3 < \overline{x} < 15.7 \}}

There is sufficient evidence at the

{5\%}

level of significance to conclude that the squirrels on the island have a different mean tail length as the species known to her

Let

{\mu}

denote the population mean tail length of the squirrel species known to the zoologist, and

{\mathrm{H}_0}

and

{\mathrm{H}_1}

be the null and alternative hypothesis respectively.

{\textrm{H}_0: \mu = 14.0}

{\textrm{H}_1: \mu \neq 14.0 \; \blacksquare}

Under

{\mathrm{H}_0,}

Z= \frac{\overline{X} - \mu}{\frac{\sigma}{\sqrt{n}}} \sim N(0,1)

For the test to not reject the null hypothesis,

\begin{gather*} -1.9600 < \frac{\overline{x}-14}{\frac{3.8}{\sqrt{20}}} < 1.9600 \\ 12.3 < \overline{x} < 15.7 \textrm{ (3 sf)} \end{gather*}

Set of values of

{\overline{x}:}

\{ \overline{x} \in \mathbb{R}: 12.3 < \overline{x} < 15.7 \} \; \blacksquare

Since

{\overline{x} = 15.8}

lies outside of the set found in (ii), the null hypothesis is rejected.

Hence there is sufficient evidence at the

{5\%}

level of significance to conclude that the squirrels on the island have a different mean tail length as the species known to her.

{\blacksquare}