At Westonci.ca, we provide clear, reliable answers to all your questions. Join our vibrant community and get the solutions you need. Our Q&A platform provides quick and trustworthy answers to your questions from experienced professionals in different areas of expertise. Experience the convenience of finding accurate answers to your questions from knowledgeable experts on our platform.
Sagot :
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.
We hope our answers were useful. Return anytime for more information and answers to any other questions you have. We appreciate your time. Please come back anytime for the latest information and answers to your questions. We're here to help at Westonci.ca. Keep visiting for the best answers to your questions.
