Question

1. in a forest that measures 5 miles by 5 miles, a sample was taken to count the number of silver maple trees in the forest. the number of trees counted in the grid is shown below. the grids where the survey was taken were chosen randomly. determine how many silver maple trees are in this forest using the random sampling technique. show your work!

Explanation:

Step1: Calculate the total area of the forest

The forest measures \(5\) miles by \(5\) miles. Using the formula for the area of a square \(A = s\times s\) (where \(s = 5\)), the total area \(A_{total}=5\times5 = 25\) square - miles.

Step2: Calculate the average number of trees per grid in the sample

The sample values are \(7\), \(3\), \(5\), \(11\), \(9\). The sum of the sample values is \(7 + 3+5 + 11+9=35\). There are \(n = 5\) grids in the sample. The average number of trees per grid in the sample \(\bar{x}=\frac{35}{5}=7\).

Step3: Assume the grid size

Assume each grid is \(1\times1\) square - mile (since the forest is divided into grids in a \(5\times5\) grid - like representation in terms of sampling).

Step4: Estimate the total number of trees

Using the formula for estimating the population from a sample (total number of trees \(N=\text{Average per unit}\times\text{Total number of units}\)). Here, the average number of trees per \(1\times1\) square - mile grid is \(7\), and the total number of \(1\times1\) square - mile grids in the \(25\) square - mile forest is \(25\). So \(N = 7\times25\).

Answer:

\(175\) silver - maple trees.