Question

Land purchase decision.A buyer for a lumber company must decide whether to buy a piece of land containing 5,000 pine trees. If 1,000 of the trees are at least 40 feet tall, the buyer will purchase the land; otherwise, he won’t. The owner of the land reports that the height of the trees has a mean of 30 feet and a standard deviation of 3 feet. Based on this information, what is the buyer’s decision?

Short answer

The buyer will not buy the land.

Step-by-step solution

Illustrating the buyers' decision

According to the buyers' condition, at least 20% of the trees should be 40 feet tall. As the mean height of the trees is 30 feet, and the standard deviation is 3 feet, the measurement of 40 feet will fall to the right of the mean.

We do not know the distribution of height of the trees, so we will use the Chebyshev rule.

$$\begin{gathered} \mathit{U}\mathit{p}\mathit{p}\mathit{e}\mathit{r}\mathit{r}\mathit{a}\mathit{n}\mathit{g}\mathit{e}\mathbf{=}\overline{\mathbf{x}}\mathbf{+}\mathbf{3}\mathit{s} \\ \mathbf{=}\mathbf{30}\mathbf{+}\mathbf{3}\mathbf{\left(}\mathbf{3}\mathbf{\right)} \\ \mathbf{=}\mathbf{30}\mathbf{+}\mathbf{9} \\ \mathbf{=}\mathbf{39} \\ \mathit{L}\mathit{o}\mathit{w}\mathit{e}\mathit{r}\mathit{r}\mathit{a}\mathit{n}\mathit{g}\mathit{e}\mathbf{=}\overline{\mathbf{x}}\mathbf{-}\mathbf{3}\mathit{s} \\ \mathbf{=}\mathbf{30}\mathbf{-}\mathbf{3}\mathbf{\left(}\mathbf{3}\mathbf{\right)} \\ \mathbf{=}\mathbf{30}\mathbf{-}\mathbf{9} \\ \mathbf{=}\mathbf{21} \end{gathered}$$

According to the rule, 88.88% of trees should fall between 21 and 39 feet.

The buyer needs at least 20% trees with 40 feet high, which is not possible as 40 lies after 39.

Therefore, the buyer will not buy the piece of land.