2018 H2 Mathematics Paper 2 Question 5

Hypothesis Testing

Sampling Theory

Answers

(i)

Since the population distribution of the MTTF is not know, the manager should take a sample of at least

{30}

fans so that the sample size is sufficiently large for the Central Limit Theorem to apply. Hence the sample mean will be normally distributed approximately and a

{Z\textrm{-}}

test can be performed.

The fans should be chosen randomly. i.e., each fan in the population should have an equal probability of being selected into the sample.

(ii)

{\textrm{H}_0: \mu = 65000}

{\textrm{H}_1: \mu < 65000}

(iii)

{\sigma^2 > 9,420,000}

Full solutions

(i)

Since the population distribution of the MTTF is not know, the manager should take a sample of at least

{30}

fans so that the sample size is sufficiently large for the Central Limit Theorem to apply. Hence the sample mean will be normally distributed approximately and a

{Z\textrm{-}}

test can be performed.

{\blacksquare}

The fans should be chosen randomly. i.e., each fan in the population should have an equal probability of being selected into the sample.

{\blacksquare}

(ii)

Let

{\mu}

denote the population mean MTTF,

{X}

denote the random variable of the MTTF of the fans,

{\sigma^2}

denote the variance, and

{\mathrm{H}_0}

and

{\mathrm{H}_1}

be the null and alternative hypothesis respectively.

{\textrm{H}_0: \mu = 65000}

{\textrm{H}_1: \mu < 65000 \; \blacksquare}

(iii)

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*} \frac{\overline{x}-\mu}{\frac{\sigma}{\sqrt{n}}} > -1.6449 \\ \frac{64230-65000}{\frac{\sigma}{\sqrt{43}}} > -1.6449 \\ -770 > -1.6449 \left( \frac{\sigma}{\sqrt{43}} \right) \\ -0.25084 \sigma < -770 \\ \sigma > 3069.7 \\ \sigma^2 > 9,420,000 \textrm{ (3 sf)} \; \blacksquare \end{gather*}