2016 H2 Mathematics Paper 2 Question 6

Hypothesis Testing

Answers

Out of syllabus

Out of syllabus

{\textrm{H}_0: \mu = 37}

{\textrm{H}_1: \mu < 37}

{\{ \overline{x} \in \mathbb{R}^+: \overline{x} < 34.8 \}}

{\{ \alpha \in \mathbb{R}^+: \alpha < 8.68 \}}

Out of syllabus

Out of syllabus

Let

{\mu}

denote the population mean age of the employees,

{X}

denote the random variable of the age of employees,

{\overline{x}}

denote the mean age of the sample of employees, and

{\mathrm{H}_0}

and

{\mathrm{H}_1}

be the null and alternative hypothesis respectively.

{\textrm{H}_0: \mu = 37}

{\textrm{H}_1: \mu < 37 \; \blacksquare}

Under

{\mathrm{H}_0,}

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

For the Managing Director's belief to be accepted (to reject the null hypothesis),

\begin{gather*} \frac{\overline{x}-37}{\frac{11.832}{\sqrt{80}}} < -1.6449 \\ \overline{x} < 34.8 \textrm{ (3 sf)} \end{gather*}

Set of values of

{\overline{x}:}

\{ \overline{x} \in \mathbb{R}: \overline{x} < 34.8 \} \; \blacksquare

Using

{\overline{x} = 35.2, }

using a GC,

p\textrm{-value} = 0.0868 \textrm{ (3 sf)}

For the Managing Director's belief to not be accepted (to not reject the null hypothesis),

\begin{align*} p\textrm{-value} &> \frac{\alpha}{100} \\ \alpha &< 8.68 \end{align*}

Set of values of

{\alpha:}

\{ \alpha \in \mathbb{R}^+: \alpha < 8.68 \} \; \blacksquare