Answer :
To solve this problem, we need to determine the linear regression equation that best fits the given data points, and then use this equation to estimate the year in which the number of new cases will reach 844. Let's go step-by-step:
### Step 1: Calculate the Linear Regression Equation
Given data points:
[tex]\[ \begin{array}{|c|c|} \hline x & y \\ \hline 0 & 1163 \\ \hline 1 & 1108 \\ \hline 2 & 1058 \\ \hline 3 & 1054 \\ \hline \end{array} \][/tex]
The linear regression equation is given by [tex]\( y = mx + c \)[/tex], where [tex]\( m \)[/tex] is the slope and [tex]\( c \)[/tex] is the y-intercept.
#### Calculate the slope ([tex]\( m \)[/tex]):
The slope formula for linear regression is:
[tex]\[ m = \frac{n(\sum xy) - (\sum x)(\sum y)}{n(\sum x^2) - (\sum x)^2} \][/tex]
Where:
- [tex]\( n \)[/tex] is the total number of data points.
- [tex]\( \sum xy \)[/tex] is the sum of the products of [tex]\( x \)[/tex] and [tex]\( y \)[/tex] values.
- [tex]\( \sum x \)[/tex] is the sum of [tex]\( x \)[/tex] values.
- [tex]\( \sum y \)[/tex] is the sum of [tex]\( y \)[/tex] values.
- [tex]\( \sum x^2 \)[/tex] is the sum of the squares of [tex]\( x \)[/tex] values.
Substitute the values from the data points:
[tex]\[ \begin{aligned} n &= 4 \\ \sum x &= 0 + 1 + 2 + 3 = 6 \\ \sum y &= 1163 + 1108 + 1058 + 1054 = 4383 \\ \sum x^2 &= 0^2 + 1^2 + 2^2 + 3^2 = 14 \\ \sum xy &= (0 \cdot 1163) + (1 \cdot 1108) + (2 \cdot 1058) + (3 \cdot 1054) = 0 + 1108 + 2116 + 3162 = 6386 \\ \end{aligned} \][/tex]
Substitute these into the slope formula:
[tex]\[ m = \frac{4(6386) - 6(4383)}{4(14) - 6^2} = \frac{25544 - 26298}{56 - 36} = \frac{-754}{20} = -37.7 \][/tex]
#### Calculate the y-intercept ([tex]\( c \)[/tex]):
The y-intercept formula for linear regression is:
[tex]\[ c = \frac{\sum y - m(\sum x)}{n} \][/tex]
Substitute [tex]\( m = -37.7 \)[/tex] and the other sums:
[tex]\[ c = \frac{4383 - (-37.7)(6)}{4} = \frac{4383 + 226.2}{4} = \frac{4609.2}{4} = 1152.3 \][/tex]
Thus, the linear regression equation is:
[tex]\[ y = -37.7x + 1152.3 \][/tex]
### Step 2: Estimate the Year When the Number of New Cases Reaches 844
We need to find [tex]\( x \)[/tex] when [tex]\( y = 844 \)[/tex].
Substitute [tex]\( y = 844 \)[/tex] into the regression equation:
[tex]\[ 844 = -37.7x + 1152.3 \][/tex]
Solving for [tex]\( x \)[/tex]:
[tex]\[ 844 - 1152.3 = -37.7x \][/tex]
[tex]\[ -308.3 = -37.7x \][/tex]
[tex]\[ x = \frac{-308.3}{-37.7} \approx 8.18 \][/tex]
Since [tex]\( x \)[/tex] represents the number of years since 2006, the estimated year can be calculated as:
[tex]\[ 2006 + 8.18 \approx 2014 \][/tex]
### Final Answers
- Regression Equation: [tex]\( y = -37.7x + 1152.3 \)[/tex]
- Estimated Year: 2014
These calculations provide the answers to the problem.
### Step 1: Calculate the Linear Regression Equation
Given data points:
[tex]\[ \begin{array}{|c|c|} \hline x & y \\ \hline 0 & 1163 \\ \hline 1 & 1108 \\ \hline 2 & 1058 \\ \hline 3 & 1054 \\ \hline \end{array} \][/tex]
The linear regression equation is given by [tex]\( y = mx + c \)[/tex], where [tex]\( m \)[/tex] is the slope and [tex]\( c \)[/tex] is the y-intercept.
#### Calculate the slope ([tex]\( m \)[/tex]):
The slope formula for linear regression is:
[tex]\[ m = \frac{n(\sum xy) - (\sum x)(\sum y)}{n(\sum x^2) - (\sum x)^2} \][/tex]
Where:
- [tex]\( n \)[/tex] is the total number of data points.
- [tex]\( \sum xy \)[/tex] is the sum of the products of [tex]\( x \)[/tex] and [tex]\( y \)[/tex] values.
- [tex]\( \sum x \)[/tex] is the sum of [tex]\( x \)[/tex] values.
- [tex]\( \sum y \)[/tex] is the sum of [tex]\( y \)[/tex] values.
- [tex]\( \sum x^2 \)[/tex] is the sum of the squares of [tex]\( x \)[/tex] values.
Substitute the values from the data points:
[tex]\[ \begin{aligned} n &= 4 \\ \sum x &= 0 + 1 + 2 + 3 = 6 \\ \sum y &= 1163 + 1108 + 1058 + 1054 = 4383 \\ \sum x^2 &= 0^2 + 1^2 + 2^2 + 3^2 = 14 \\ \sum xy &= (0 \cdot 1163) + (1 \cdot 1108) + (2 \cdot 1058) + (3 \cdot 1054) = 0 + 1108 + 2116 + 3162 = 6386 \\ \end{aligned} \][/tex]
Substitute these into the slope formula:
[tex]\[ m = \frac{4(6386) - 6(4383)}{4(14) - 6^2} = \frac{25544 - 26298}{56 - 36} = \frac{-754}{20} = -37.7 \][/tex]
#### Calculate the y-intercept ([tex]\( c \)[/tex]):
The y-intercept formula for linear regression is:
[tex]\[ c = \frac{\sum y - m(\sum x)}{n} \][/tex]
Substitute [tex]\( m = -37.7 \)[/tex] and the other sums:
[tex]\[ c = \frac{4383 - (-37.7)(6)}{4} = \frac{4383 + 226.2}{4} = \frac{4609.2}{4} = 1152.3 \][/tex]
Thus, the linear regression equation is:
[tex]\[ y = -37.7x + 1152.3 \][/tex]
### Step 2: Estimate the Year When the Number of New Cases Reaches 844
We need to find [tex]\( x \)[/tex] when [tex]\( y = 844 \)[/tex].
Substitute [tex]\( y = 844 \)[/tex] into the regression equation:
[tex]\[ 844 = -37.7x + 1152.3 \][/tex]
Solving for [tex]\( x \)[/tex]:
[tex]\[ 844 - 1152.3 = -37.7x \][/tex]
[tex]\[ -308.3 = -37.7x \][/tex]
[tex]\[ x = \frac{-308.3}{-37.7} \approx 8.18 \][/tex]
Since [tex]\( x \)[/tex] represents the number of years since 2006, the estimated year can be calculated as:
[tex]\[ 2006 + 8.18 \approx 2014 \][/tex]
### Final Answers
- Regression Equation: [tex]\( y = -37.7x + 1152.3 \)[/tex]
- Estimated Year: 2014
These calculations provide the answers to the problem.
