Question

the dot plots display the height, rounded to the nearest foot, of maple trees from two different tree farms.

1. compare the mean and standard deviation of the two data sets.

the means of the two data sets both equal feet. the standard deviation of the first data set is than the standard deviation of the second data set.

1. what does the standard deviation tell you about the trees at these farms?

at the second farm, there is a range of heights than at the first farm, where the tree heights are one another.

Explanation:

Step1: Calculate the mean of the first data - set

Let the data - points be \(x_i\) and their frequencies be \(f_i\). The formula for the mean \(\bar{x}=\frac{\sum_{i = 1}^{n}x_if_i}{\sum_{i = 1}^{n}f_i}\). For the first data - set:
\[

$$\begin{align*} &(6\times1)+(7\times2)+(8\times3)+(9\times2)+(10\times1)\\ =&6 + 14+24 + 18+10\\ =&72 \end{align*}$$

\]
The total number of data - points \(n=1 + 2+3 + 2+1=9\). So the mean \(\bar{x}_1=\frac{72}{9}=8\) feet.
For the second data - set:
\[

$$\begin{align*} &(4\times1)+(6\times2)+(8\times3)+(10\times2)+(12\times1)\\ =&4+12 + 24+20+12\\ =&72 \end{align*}$$

\]
The total number of data - points \(n = 1+2 + 3+2+1=9\). So the mean \(\bar{x}_2=\frac{72}{9}=8\) feet.

Step2: Calculate the standard deviation conceptually

The standard deviation \(\sigma=\sqrt{\frac{\sum_{i = 1}^{n}(x_i-\bar{x})^2f_i}{n}}\). Looking at the dot - plots, the first data - set has data points more clustered around the mean (8). The second data - set has data points more spread out (from 4 to 12 compared to 6 to 10 in the first data - set). So the standard deviation of the first data - set is less than the standard deviation of the second data - set.

Step3: Interpret the standard deviation

The standard deviation measures the amount of variation or dispersion in a set of data. A larger standard deviation in the second farm means there is a larger range of heights. In the first farm, since the standard deviation is smaller, the tree heights are more similar to one another.

Answer:

1. The means of the two data sets both equal 8 feet. The standard deviation of the first data set is less than the standard deviation of the second data set.
2. At the second farm, there is a larger range of heights than at the first farm, where the tree heights are more similar to one another.