一台自动分装机被设定为向杯中注入热巧克力。根据长期观察，发现每杯热巧克力的容量（以 ml 为单位）符合均值为 55、标准差为 10 的分布。 抽取了一个包含 $n$ 杯热巧克力的大型随机样本。已知样本均值超过 54 ml 的概率大于 0.77，求 $n$ 的最小值。

1. State the approximate distribution of the sample mean using the Central Limit Theorem.

Let $X$ be the volume of hot chocolate in a cup. We are given the population mean $\mu = 55$ and standard deviation $\sigma = 10$.
Since the sample size $n$ is large, by the Central Limit Theorem, the sample mean $\bar{X}$ is approximately normally distributed:
$\bar{X} \sim \mathrm{N}\left(55, \frac{10^2}{n}\right) \text{ approximately.}$

2. Standardise the probability statement to form an inequality involving $Z$.

We are given that $\mathrm{P}(\bar{X} > 54) > 0.77$.
Standardising the inequality to the standard normal variable $Z$:
$\mathrm{P}\left(Z > \frac{54 - 55}{10/\sqrt{n}}\right) > 0.77$
$\mathrm{P}\left(Z > -\frac{\sqrt{n}}{10}\right) > 0.77$

3. Solve for the least integer value of $n$ using the inverse normal distribution.

By symmetry of the normal distribution, $\mathrm{P}\left(Z < \frac{\sqrt{n}}{10}\right) > 0.77$.
Using a graphing calculator to find the inverse normal value for an area of $0.77$:
$z \approx 0.73885$
Thus, we require:
$\frac{\sqrt{n}}{10} > 0.73885$
$\sqrt{n} > 7.3885$
$n > 54.59$
Since $n$ must be an integer, the least value of $n$ is $55$.