Certainly! Let's go through the problem step by step to determine the annual trend in tornado counts.
### Step 1: Organize Data
First, we have the following data for the number of tornadoes over different years:
[tex]\[
\begin{array}{|c|c|}
\hline
\text{Year} & \text{Number of Tornadoes} \\
\hline
1350 & 200 \\
\hline
1360 & 600 \\
\hline
1370 & 660 \\
\hline
1300 & 180 \\
\hline
1000 & 1150 \\
\hline
2000 & 1350 \\
\hline
2010 & 1300 \\
\hline
\end{array}
\][/tex]
### Step 2: Determine the Trend
To find the trend in the number of tornadoes over the years, we use linear regression. In linear regression, we fit a line of the form [tex]\( y = mx + c \)[/tex], where [tex]\( m \)[/tex] is the slope and [tex]\( c \)[/tex] is the intercept.
- Slope (m): Indicates the rate of change in the number of tornadoes per year.
- Intercept (c): Represents the baseline number of tornadoes when the year is zero.
Based on the calculations, we have obtained the following parameters:
- Slope ([tex]\( m \)[/tex]): [tex]\( 0.7146 \)[/tex]
- Intercept ([tex]\( c \)[/tex]): [tex]\( -283.5417 \)[/tex]
### Step 3: Trend Interpretation
The slope and intercept allow us to predict the number of tornadoes for any given year. The linear equation based on the slope and intercept is:
[tex]\[ \text{Number of Tornadoes} = 0.7146 \times (\text{Year}) - 283.5417 \][/tex]
### Step 4: Predicted Tornado Counts
Using the above linear equation, we compute the predicted tornado counts for each given year:
1. For year 1350:
[tex]\[ \text{Predicted Tornadoes} = 0.7146 \times 1350 - 283.5417 = 681.18 \][/tex]
2. For year 1360:
[tex]\[ \text{Predicted Tornadoes} = 0.7146 \times 1360 - 283.5417 = 688.33 \][/tex]
3. For year 1370:
[tex]\[ \text{Predicted Tornadoes} = 0.7146 \times 1370 - 283.5417 = 695.47 \][/tex]
4. For year 1300:
[tex]\[ \text{Predicted Tornadoes} = 0.7146 \times 1300 - 283.5417 = 645.45 \][/tex]
5. For year 1000:
[tex]\[ \text{Predicted Tornadoes} = 0.7146 \times 1000 - 283.5417 = 431.07 \][/tex]
6. For year 2000:
[tex]\[ \text{Predicted Tornadoes} = 0.7146 \times 2000 - 283.5417 = 1145.68 \][/tex]
7. For year 2010:
[tex]\[ \text{Predicted Tornadoes} = 0.7146 \times 2010 - 283.5417 = 1152.82 \][/tex]
### Conclusion
The linear equation [tex]\( 0.7146 \times \text{Year} - 283.5417 \)[/tex] provides a way to predict the number of tornadoes for any given year. The trend indicates that the number of tornadoes increases at a rate of approximately [tex]\( 0.7146 \)[/tex] tornadoes per year.
Here are the predicted tornado counts for the specified years:
[tex]\[
\begin{array}{|c|c|}
\hline
\text{Year} & \text{Predicted Tornadoes} \\
\hline
1350 & 681.18 \\
\hline
1360 & 688.33 \\
\hline
1370 & 695.47 \\
\hline
1300 & 645.45 \\
\hline
1000 & 431.07 \\
\hline
2000 & 1145.68 \\
\hline
2010 & 1152.82 \\
\hline
\end{array}
\][/tex]
Understanding the trend helps predict future occurrences and prepare accordingly.
