Answer :
Sure! Let's go step by step to estimate the calendar year in which the profit will reach 248 thousand dollars.
### Step 1: Organize the Data
Given data points in the problem:
| Years since 2011 ([tex]$x$[/tex]) | Profits ([tex]$y$[/tex]) (in thousands of dollars) |
|------------------------|-----------------------------------------|
| 0 | 95 |
| 1 | 133 |
| 2 | 131 |
| 3 | 163 |
### Step 2: Perform Linear Regression
To find the relationship between the number of years since 2011 ([tex]$x$[/tex]) and profits ([tex]$y$[/tex]), we will use linear regression. This will give us an equation of the form:
[tex]\[ y = mx + b \][/tex]
where:
- [tex]$m$[/tex] is the slope,
- [tex]$b$[/tex] is the y-intercept.
From the provided data, the linear regression yields:
[tex]\[ m = 20.20 \][/tex]
[tex]\[ b = 100.20 \][/tex]
Thus, the regression equation is:
[tex]\[ y = 20.20x + 100.20 \][/tex]
### Step 3: Set the Target Profit
We are asked to estimate the calendar year in which the profits will reach 248 thousand dollars. So, we set:
[tex]\[ y = 248 \][/tex]
### Step 4: Solve for [tex]$x$[/tex]
Using the regression equation:
[tex]\[ 248 = 20.20x + 100.20 \][/tex]
We solve for [tex]$x$[/tex] as follows:
[tex]\[ 248 - 100.20 = 20.20x \][/tex]
[tex]\[ 147.80 = 20.20x \][/tex]
[tex]\[ x = \frac{147.80}{20.20} \][/tex]
[tex]\[ x \approx 7.32 \][/tex]
This means that it will take approximately 7.32 years since 2011 for the profit to reach 248 thousand dollars.
### Step 5: Calculate the Calendar Year
To find the actual calendar year, we add the number of years since 2011 to the year 2011:
[tex]\[ \text{Calendar Year} = 2011 + 7.32 \][/tex]
[tex]\[ \text{Calendar Year} \approx 2018.32 \][/tex]
Thus, the profit is projected to reach 248 thousand dollars in the calendar year around mid-2018, which is represented as 2018.32.
### Summary:
- Regression Equation: [tex]\[ y = 20.20x + 100.20 \][/tex]
- Estimated Calendar Year: [tex]\[ \approx 2018.32 \][/tex]
Therefore, the company is projected to reach a profit of 248 thousand dollars around mid-2018.
### Step 1: Organize the Data
Given data points in the problem:
| Years since 2011 ([tex]$x$[/tex]) | Profits ([tex]$y$[/tex]) (in thousands of dollars) |
|------------------------|-----------------------------------------|
| 0 | 95 |
| 1 | 133 |
| 2 | 131 |
| 3 | 163 |
### Step 2: Perform Linear Regression
To find the relationship between the number of years since 2011 ([tex]$x$[/tex]) and profits ([tex]$y$[/tex]), we will use linear regression. This will give us an equation of the form:
[tex]\[ y = mx + b \][/tex]
where:
- [tex]$m$[/tex] is the slope,
- [tex]$b$[/tex] is the y-intercept.
From the provided data, the linear regression yields:
[tex]\[ m = 20.20 \][/tex]
[tex]\[ b = 100.20 \][/tex]
Thus, the regression equation is:
[tex]\[ y = 20.20x + 100.20 \][/tex]
### Step 3: Set the Target Profit
We are asked to estimate the calendar year in which the profits will reach 248 thousand dollars. So, we set:
[tex]\[ y = 248 \][/tex]
### Step 4: Solve for [tex]$x$[/tex]
Using the regression equation:
[tex]\[ 248 = 20.20x + 100.20 \][/tex]
We solve for [tex]$x$[/tex] as follows:
[tex]\[ 248 - 100.20 = 20.20x \][/tex]
[tex]\[ 147.80 = 20.20x \][/tex]
[tex]\[ x = \frac{147.80}{20.20} \][/tex]
[tex]\[ x \approx 7.32 \][/tex]
This means that it will take approximately 7.32 years since 2011 for the profit to reach 248 thousand dollars.
### Step 5: Calculate the Calendar Year
To find the actual calendar year, we add the number of years since 2011 to the year 2011:
[tex]\[ \text{Calendar Year} = 2011 + 7.32 \][/tex]
[tex]\[ \text{Calendar Year} \approx 2018.32 \][/tex]
Thus, the profit is projected to reach 248 thousand dollars in the calendar year around mid-2018, which is represented as 2018.32.
### Summary:
- Regression Equation: [tex]\[ y = 20.20x + 100.20 \][/tex]
- Estimated Calendar Year: [tex]\[ \approx 2018.32 \][/tex]
Therefore, the company is projected to reach a profit of 248 thousand dollars around mid-2018.
