A botanist is studying the height of a rare plant species. Out of a population of $150$ plants, the mean height is $32.4\text{ cm}$ with a standard deviation of $8.1\text{ cm}$. If a random sample of $40$ plants is chosen, what is the probability that the sample mean height is between $30\text{ cm}$ and $35\text{ cm}$? Should the finite correction factor be used?