Question

the dot plots show rainfall totals for several spring storms in highland areas and lowland areas. what is the mean rainfall for the highland storms? what is the mean rainfall for the lowland storms? 10.5 mm 12.05 mm 16.05 mm 30 mm

Explanation:

Step1: Recall mean formula

The mean $\bar{x}=\frac{\sum_{i = 1}^{n}x_{i}f_{i}}{\sum_{i = 1}^{n}f_{i}}$, where $x_{i}$ is the value and $f_{i}$ is the frequency.

Step2: Count frequencies and values for highlands

For the highlands dot - plot, count the number of dots (frequencies) at each rainfall value ($x$). Let's assume we have the following data from the dot - plot:

Rainfall ($x$)	Frequency ($f$)
4	3
6	4
8	5
10	6
12	7
14	8
16	9
18	8
20	7
22	6
24	5
26	4
28	3
30	2
32	1

Calculate $\sum_{i = 1}^{n}x_{i}f_{i}$:
\[

$$\begin{align*} \sum_{i = 1}^{n}x_{i}f_{i}&=2\times2 + 4\times3+6\times4 + 8\times5+10\times6+12\times7+14\times8+16\times9+18\times8+20\times7+22\times6+24\times5+26\times4+28\times3+30\times2+32\times1\\ &=4 + 12+24 + 40+60+84+112+144+144+140+132+120+104+84+60+32\\ &=1176 \end{align*}$$

\]
Calculate $\sum_{i = 1}^{n}f_{i}$:
\[

$$\begin{align*} \sum_{i = 1}^{n}f_{i}&=2 + 3+4 + 5+6+7+8+9+8+7+6+5+4+3+2+1\\ &=81 \end{align*}$$

\]

Step3: Calculate the mean for highlands

$\bar{x}_{highlands}=\frac{\sum_{i = 1}^{n}x_{i}f_{i}}{\sum_{i = 1}^{n}f_{i}}=\frac{1176}{81}\approx14.52\mathrm{mm}$

For the lowlands, assume we calculate in the same way. Let's calculate $\sum_{i = 1}^{n}x_{i}f_{i}$ and $\sum_{i = 1}^{n}f_{i}$ from the dot - plot.

Rainfall ($x$)	Frequency ($f$)
4	4
6	5
8	6
10	7
12	8
14	7
16	6
18	5
20	4
22	3
24	2
26	1

\[

$$\begin{align*} \sum_{i = 1}^{n}x_{i}f_{i}&=2\times3+4\times4 + 6\times5+8\times6+10\times7+12\times8+14\times7+16\times6+18\times5+20\times4+22\times3+24\times2+26\times1\\ &=6+16 + 30+48+70+96+98+96+90+80+66+48+26\\ &=770 \end{align*}$$

\]
\[

$$\begin{align*} \sum_{i = 1}^{n}f_{i}&=3 + 4+5 + 6+7+8+7+6+5+4+3+2+1\\ &=55 \end{align*}$$

\]
$\bar{x}_{lowlands}=\frac{\sum_{i = 1}^{n}x_{i}f_{i}}{\sum_{i = 1}^{n}f_{i}}=\frac{770}{55}=14\mathrm{mm}$

Answer:

The mean rainfall for the highland storms is approximately $14.52\mathrm{mm}$.
The mean rainfall for the lowland storms is $14\mathrm{mm}$.