Solution — 2022 Question 10 Sampling| Hypothesis Testing

(a)

$μ$ is the population mean checking time (in seconds) of a metal disc $∎$

$H_{0} : μ = 16.2$
$H_{1} : μ \neq 16.2$

$\begin{matrix} Unbiased estimate of population mean \\ = \overline{t} \\ = \frac{\sum t}{n} \\ = \frac{645.6}{40} \\ = 16.14 \end{matrix}$

$\begin{matrix} Unbiased estimate of population variance \\ = s^{2} \\ = \frac{1}{n - 1} (\sum t^{2} - \frac{{(\sum t)}^{2}}{n}) \\ = \frac{1}{40 - 1} (10 421 - \frac{{(645.6)}^{2}}{40}) \\ = 0.026 051 \end{matrix}$

Under $H_{0},$ at $α %$ level of significance, $ ParseError: Can't use function '$' in math mode at position 1: $̲ \bar{T} \sim \mathrm{N} \left( 16.2, \frac{0.026,051}{40} \right) $$ Z = \frac{\bar{T}-16.2}{\sqrt{\frac{0.026,051}{40}}} \sim \mathrm{N}(0,1) approximately by the Central Limit Theorem since $ is large

From GC, $p -value = 0.018 719$

Since there is insufficient evidence to reject $H_{0},$

\begin{align*} p\text{-value} &> \alpha \% \\ 0.018\,719 &> \frac{\alpha }{100} \\ 1.8719 &> \alpha \\ \alpha &< 1.8719 \\ \alpha &< 1.87\; \QED \text{ (to 3 s.f.)} \end{align*}$$ ParseError: {align*} can be used only in display mode.

(b)

Let $X$ denote the random variable of the length (in cm) of a metal rod
Let $μ$ denote the population mean length (in cm) of a metal rod

$H_{0} : μ = 12$
$H_{1} : μ < 12$

From the question, $\overline{x} = 11.8, σ^{2} = 1.1, n = 60$

Under $H_{0},$ at $0.05 %$ level of significance, $\overline{X} \sim N (12, \frac{1.1}{60})$ $Z = \frac{\overline{X} - 12}{\sqrt{\frac{1.1}{60}}} \sim N (0,1)$ approximately by the Central Limit Theorem since $n = 60$ is large

From GC,
$p$ -value = $0.069825 > 0.0005 \Rightarrow Do not reject H_{0}$

Hence there is insufficient evidence at the $0.05 %$ level of significance to conclude whether the mean length of the rods is less than 12 cm $∎$