An automatic dispensing machine is set to dispense hot chocolate into cups. Based on observations over a long period of time, the cups were found to contain volumes of hot chocolate, in ml, with mean 55 and standard deviation 10. A large random sample of n cups of hot chocolate is taken. Given that the probability that the sample mean exceeds 54 ml is more than 0.77, find the least value of n.

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 $55$.