IDNLearn.com is your trusted platform for finding reliable answers. Our community provides timely and precise responses to help you understand and solve any issue you face.
Sagot :
Let's go through the steps to solve the problem step-by-step.
### Step 1: Organize the Data
We have the following data points:
[tex]\[ \begin{array}{|c|c|} \hline x & y \\ \hline 0 & 1047 \\ \hline 1 & 1020 \\ \hline 2 & 989 \\ \hline 3 & 943 \\ \hline 4 & 964 \\ \hline 5 & 880 \\ \hline \end{array} \][/tex]
### Step 2: Calculate the Means
First, we calculate the mean of x and y:
[tex]\[ \bar{x} = \frac{0 + 1 + 2 + 3 + 4 + 5}{6} = \frac{15}{6} = 2.5 \][/tex]
[tex]\[ \bar{y} = \frac{1047 + 1020 + 989 + 943 + 964 + 880}{6} = \frac{5843}{6} \approx 973.8 \][/tex]
### Step 3: Calculate the Slope (m) and y-Intercept (b)
The formula for the slope [tex]\( m \)[/tex] of the least squares regression line is:
[tex]\[ m = \frac{\sum{(x_i - \bar{x})(y_i - \bar{y})}}{\sum{(x_i - \bar{x})^2}} \][/tex]
Let's calculate each part step-by-step.
#### Numerator Calculation:
[tex]\[ \sum{(x_i - \bar{x})(y_i - \bar{y})} \][/tex]
[tex]\[ = (0 - 2.5)(1047 - 973.8) + (1 - 2.5)(1020 - 973.8) + (2 - 2.5)(989 - 973.8) + (3 - 2.5)(943 - 973.8) + (4 - 2.5)(964 - 973.8) + (5 - 2.5)(880 - 973.8) \][/tex]
[tex]\[ = (-2.5 \times 73.2) + (-1.5 \times 46.2) + (-0.5 \times 15.2) + (0.5 \times (-30.8)) + (1.5 \times (-9.8)) + (2.5 \times (-93.8)) \][/tex]
[tex]\[ = -183 + -69.3 + -7.6 + -15.4 + -14.7 + -234.5 = -524.5 \][/tex]
#### Denominator Calculation:
[tex]\[ \sum{(x_i - \bar{x})^2} = (0 - 2.5)^2 + (1 - 2.5)^2 + (2 - 2.5)^2 + (3 - 2.5)^2 + (4 - 2.5)^2 + (5 - 2.5)^2 \][/tex]
[tex]\[ = 6.25 + 2.25 + 0.25 + 0.25 + 2.25 + 6.25 = 17.5 \][/tex]
#### Slope (m):
[tex]\[ m = \frac{-524.5}{17.5} \approx -30 \][/tex]
#### y-Intercept (b):
[tex]\[ b = \bar{y} - m\bar{x} \][/tex]
[tex]\[ b = 973.8 - (-30 \times 2.5) = 973.8 + 75 = 1048.8 \][/tex]
### Step 4: Write the Linear Regression Equation
The linear regression equation is:
[tex]\[ y = mx + b \][/tex]
Substituting the calculated [tex]\( m \)[/tex] and [tex]\( b \)[/tex]:
[tex]\[ y = -30x + 1048.8 \][/tex]
Round the coefficients to the nearest tenth:
[tex]\[ y = -30x + 1048.8 \][/tex]
### Step 5: Project the Number of New Cases for 2026
To find the number of new cases for 2026, we need to calculate [tex]\( x \)[/tex] for the year 2026.
Since [tex]\( x \)[/tex] represents the number of years since 2014:
[tex]\[ x = 2026 - 2014 = 12 \][/tex]
Substitute [tex]\( x = 12 \)[/tex] into the linear regression equation:
[tex]\[ y = -30(12) + 1048.8 \][/tex]
[tex]\[ y = -360 + 1048.8 \][/tex]
[tex]\[ y = 688.8 \][/tex]
Rounding to the nearest whole number, the projected number of new cases for 2026 is:
[tex]\[ \boxed{689} \][/tex]
### Step 1: Organize the Data
We have the following data points:
[tex]\[ \begin{array}{|c|c|} \hline x & y \\ \hline 0 & 1047 \\ \hline 1 & 1020 \\ \hline 2 & 989 \\ \hline 3 & 943 \\ \hline 4 & 964 \\ \hline 5 & 880 \\ \hline \end{array} \][/tex]
### Step 2: Calculate the Means
First, we calculate the mean of x and y:
[tex]\[ \bar{x} = \frac{0 + 1 + 2 + 3 + 4 + 5}{6} = \frac{15}{6} = 2.5 \][/tex]
[tex]\[ \bar{y} = \frac{1047 + 1020 + 989 + 943 + 964 + 880}{6} = \frac{5843}{6} \approx 973.8 \][/tex]
### Step 3: Calculate the Slope (m) and y-Intercept (b)
The formula for the slope [tex]\( m \)[/tex] of the least squares regression line is:
[tex]\[ m = \frac{\sum{(x_i - \bar{x})(y_i - \bar{y})}}{\sum{(x_i - \bar{x})^2}} \][/tex]
Let's calculate each part step-by-step.
#### Numerator Calculation:
[tex]\[ \sum{(x_i - \bar{x})(y_i - \bar{y})} \][/tex]
[tex]\[ = (0 - 2.5)(1047 - 973.8) + (1 - 2.5)(1020 - 973.8) + (2 - 2.5)(989 - 973.8) + (3 - 2.5)(943 - 973.8) + (4 - 2.5)(964 - 973.8) + (5 - 2.5)(880 - 973.8) \][/tex]
[tex]\[ = (-2.5 \times 73.2) + (-1.5 \times 46.2) + (-0.5 \times 15.2) + (0.5 \times (-30.8)) + (1.5 \times (-9.8)) + (2.5 \times (-93.8)) \][/tex]
[tex]\[ = -183 + -69.3 + -7.6 + -15.4 + -14.7 + -234.5 = -524.5 \][/tex]
#### Denominator Calculation:
[tex]\[ \sum{(x_i - \bar{x})^2} = (0 - 2.5)^2 + (1 - 2.5)^2 + (2 - 2.5)^2 + (3 - 2.5)^2 + (4 - 2.5)^2 + (5 - 2.5)^2 \][/tex]
[tex]\[ = 6.25 + 2.25 + 0.25 + 0.25 + 2.25 + 6.25 = 17.5 \][/tex]
#### Slope (m):
[tex]\[ m = \frac{-524.5}{17.5} \approx -30 \][/tex]
#### y-Intercept (b):
[tex]\[ b = \bar{y} - m\bar{x} \][/tex]
[tex]\[ b = 973.8 - (-30 \times 2.5) = 973.8 + 75 = 1048.8 \][/tex]
### Step 4: Write the Linear Regression Equation
The linear regression equation is:
[tex]\[ y = mx + b \][/tex]
Substituting the calculated [tex]\( m \)[/tex] and [tex]\( b \)[/tex]:
[tex]\[ y = -30x + 1048.8 \][/tex]
Round the coefficients to the nearest tenth:
[tex]\[ y = -30x + 1048.8 \][/tex]
### Step 5: Project the Number of New Cases for 2026
To find the number of new cases for 2026, we need to calculate [tex]\( x \)[/tex] for the year 2026.
Since [tex]\( x \)[/tex] represents the number of years since 2014:
[tex]\[ x = 2026 - 2014 = 12 \][/tex]
Substitute [tex]\( x = 12 \)[/tex] into the linear regression equation:
[tex]\[ y = -30(12) + 1048.8 \][/tex]
[tex]\[ y = -360 + 1048.8 \][/tex]
[tex]\[ y = 688.8 \][/tex]
Rounding to the nearest whole number, the projected number of new cases for 2026 is:
[tex]\[ \boxed{689} \][/tex]
Thank you for contributing to our discussion. Don't forget to check back for new answers. Keep asking, answering, and sharing useful information. Find reliable answers at IDNLearn.com. Thanks for stopping by, and come back for more trustworthy solutions.
