Answer :
Certainly! Let's break down the solution into two parts: fitting the straight-line trend and estimating the production for 2019.
### 1. Fit the Straight Line Trend
To fit a straight-line trend to the given data, we need to determine the equation of the line in the form:
[tex]\[ y = mx + b \][/tex]
where:
- [tex]\( y \)[/tex] is the production (in million tonnes),
- [tex]\( x \)[/tex] is the year,
- [tex]\( m \)[/tex] is the slope of the line,
- [tex]\( b \)[/tex] is the intercept.
Given data:
- Years ([tex]\( x \)[/tex]): 2012, 2013, 2014, 2015, 2016, 2017, 2018
- Production ([tex]\( y \)[/tex]): 80, 90, 92, 83, 94, 99, 92
Using statistical methods, we determine:
- Slope ([tex]\( m \)[/tex]): 2.0
- Intercept ([tex]\( b \)[/tex]): -3940.0
The equation of our trend line will be:
[tex]\[ y = 2.0x - 3940.0 \][/tex]
Now we calculate the trend values for each given year by plugging the year values into the trend equation.
For 2012:
[tex]\[ y = 2.0 \times 2012 - 3940.0 = 84.0 \][/tex]
For 2013:
[tex]\[ y = 2.0 \times 2013 - 3940.0 = 86.0 \][/tex]
For 2014:
[tex]\[ y = 2.0 \times 2014 - 3940.0 = 88.0 \][/tex]
For 2015:
[tex]\[ y = 2.0 \times 2015 - 3940.0 = 90.0 \][/tex]
For 2016:
[tex]\[ y = 2.0 \times 2016 - 3940.0 = 92.0 \][/tex]
For 2017:
[tex]\[ y = 2.0 \times 2017 - 3940.0 = 94.0 \][/tex]
For 2018:
[tex]\[ y = 2.0 \times 2018 - 3940.0 = 96.0 \][/tex]
Thus, the trend values are:
[tex]\[ [84.0, 86.0, 88.0, 90.0, 92.0, 94.0, 96.0] \][/tex]
### 2. Estimate the Likely Sales During 2019
To estimate the likely production for 2019, we use the same trend equation and substitute [tex]\( x = 2019 \)[/tex]:
[tex]\[ y = 2.0 \times 2019 - 3940.0 \][/tex]
Calculating this:
[tex]\[ y = 2.0 \times 2019 - 3940.0 = 98.0 \][/tex]
Therefore, the estimated production for the year 2019 is [tex]\( 98.0 \)[/tex] million tonnes.
### Summary
1. The fitted straight-line trend equation is:
[tex]\[ y = 2.0x - 3940.0 \][/tex]
2. The trend values for the given years are:
[tex]\[ [84.0, 86.0, 88.0, 90.0, 92.0, 94.0, 96.0] \][/tex]
3. The estimated production for the year 2019 is:
[tex]\[ 98.0 \][/tex] million tonnes.
### 1. Fit the Straight Line Trend
To fit a straight-line trend to the given data, we need to determine the equation of the line in the form:
[tex]\[ y = mx + b \][/tex]
where:
- [tex]\( y \)[/tex] is the production (in million tonnes),
- [tex]\( x \)[/tex] is the year,
- [tex]\( m \)[/tex] is the slope of the line,
- [tex]\( b \)[/tex] is the intercept.
Given data:
- Years ([tex]\( x \)[/tex]): 2012, 2013, 2014, 2015, 2016, 2017, 2018
- Production ([tex]\( y \)[/tex]): 80, 90, 92, 83, 94, 99, 92
Using statistical methods, we determine:
- Slope ([tex]\( m \)[/tex]): 2.0
- Intercept ([tex]\( b \)[/tex]): -3940.0
The equation of our trend line will be:
[tex]\[ y = 2.0x - 3940.0 \][/tex]
Now we calculate the trend values for each given year by plugging the year values into the trend equation.
For 2012:
[tex]\[ y = 2.0 \times 2012 - 3940.0 = 84.0 \][/tex]
For 2013:
[tex]\[ y = 2.0 \times 2013 - 3940.0 = 86.0 \][/tex]
For 2014:
[tex]\[ y = 2.0 \times 2014 - 3940.0 = 88.0 \][/tex]
For 2015:
[tex]\[ y = 2.0 \times 2015 - 3940.0 = 90.0 \][/tex]
For 2016:
[tex]\[ y = 2.0 \times 2016 - 3940.0 = 92.0 \][/tex]
For 2017:
[tex]\[ y = 2.0 \times 2017 - 3940.0 = 94.0 \][/tex]
For 2018:
[tex]\[ y = 2.0 \times 2018 - 3940.0 = 96.0 \][/tex]
Thus, the trend values are:
[tex]\[ [84.0, 86.0, 88.0, 90.0, 92.0, 94.0, 96.0] \][/tex]
### 2. Estimate the Likely Sales During 2019
To estimate the likely production for 2019, we use the same trend equation and substitute [tex]\( x = 2019 \)[/tex]:
[tex]\[ y = 2.0 \times 2019 - 3940.0 \][/tex]
Calculating this:
[tex]\[ y = 2.0 \times 2019 - 3940.0 = 98.0 \][/tex]
Therefore, the estimated production for the year 2019 is [tex]\( 98.0 \)[/tex] million tonnes.
### Summary
1. The fitted straight-line trend equation is:
[tex]\[ y = 2.0x - 3940.0 \][/tex]
2. The trend values for the given years are:
[tex]\[ [84.0, 86.0, 88.0, 90.0, 92.0, 94.0, 96.0] \][/tex]
3. The estimated production for the year 2019 is:
[tex]\[ 98.0 \][/tex] million tonnes.
