一家工厂生产长方形白色大理石地板砖。地板砖的长度（单位：厘米）可以看作是一个均值为 18.9、标准差为 0.3 的正态随机变量。地板砖的宽度（单位：厘米）可以看作是一个独立的正态随机变量，均值为 8.9，标准差为 0.1。求随机抽取的十块地板砖的总周长的均值和标准差。

1. Define the random variable and state the exact binomial distribution with its mean and variance.

Let $X$ be the number of red-tinted tiles in a packaging of 500 tiles.
$X \sim \text{B}(500, 0.6)$
Mean, $\mu = np = 500(0.6) = 300$
Variance, $\sigma^2 = np(1-p) = 300(0.4) = 120$

2. State the normal approximation and set up the probability inequality with continuity correction.

Since $n$ is large and both $np = 300 > 5$ and $n(1-p) = 200 > 5$, $X$ can be suitably approximated by a normal distribution:
$Y \sim \text{N}(300, 120)$
We require the claim to be correct at least 95% of the time, so $P(X \ge k) \ge 0.95$.
Applying continuity correction for discrete to continuous approximation:
$P(Y \ge k - 0.5) \ge 0.95$

3. Standardise the normal variable and use the inverse normal distribution to find the range of values for k.

Standardising the random variable:
$P\left(Z \ge \frac{k - 0.5 - 300}{\sqrt{120}}\right) \ge 0.95$
From the standard normal distribution, the critical $z$-value for an upper tail area of $0.95$ (or lower tail of $0.05$) is approximately $-1.6449$.
Thus, we require:
$\frac{k - 300.5}{\sqrt{120}} \le -1.6449$
$k - 300.5 \le -1.6449\sqrt{120}$
$k \le 300.5 - 18.018$
$k \le 282.48$

4. Conclude the final greatest integer value for k.

Since $k$ represents a number of tiles, it must be an integer.
The greatest possible integer value satisfying $k \le 282.48$ is $k = 282$.