Solution — 2025 Question 13 Binomial Distribution| Sampling

(a)

$\begin{matrix} Unbiased estimate of population mean \\ = \overline{x} \\ = \frac{\sum (x - 60)}{n} + 60 \\ = \frac{72}{50} + 60 \\ = \frac{1536}{25} \\ = 61.44 ∎ \end{matrix}$

$\begin{matrix} Unbiased estimate of population variance \\ = s^{2} \\ = \frac{1}{n - 1} (\sum (x - 60)^{2} - \frac{\sum (x - 60)^{2}}{n}) \\ = \frac{1}{49} (758 - \frac{72^{2}}{50}) \\ = \frac{16 358}{1225} \\ = 13.4 ∎ (to 3 s.f.) \end{matrix}$

(b)

Let $\overline{X}$ be the mean mass (in grams) of a randomly chosen tray of $36$ eggs.

By the Central Limit Theorem, since $n = 36$ is large,

$ ParseError: Can't use function '$' in math mode at position 1: $̲ \bar{X} \sim \mathrm{N} \left( 61.44, \frac{13.353}{36} \right) \text{ approximately}

\begin{align*} \mathrm{P}\left(61 < \bar{X} < 63\right) &= 0.759\,78 \\ &= 0.760\; \QED \text{ (to 3 s.f.)} \end{align*}$$ ParseError: {align*} can be used only in display mode.

[finding \mathrm{P}\left(61 < \bar{X} < 63\right)]

(c)

Let $T$ represent the number of trays (out of 10) where the mean mass of an egg lies between 61 g and 63 g.

$ ParseError: Can't use function '$' in math mode at position 1: $̲ T \sim \mathrm{B} \left( 10, 0.759,78 \right)

\begin{align*} \mathrm{P}\left(T \geq 7\right) &= 1 - \mathrm{P}\left(T \leq 6\right) \\ &= 1 - 0.201\,75 \\ &= 0.798\,25 \\ &= 0.798\; \QED \text{ (to 3 s.f.)} \end{align*}$$ ParseError: {align*} can be used only in display mode.

[probability of T \geq 7]