Question
Question Use the table to answer the question. x y 1 5 2 9 3 10 4 10.5 5 11 Use a spreadsheet calculator to determine the equation of the line of best fit for the data in this table. Round to the nearest hundredth, if needed. (1 point) Responses y=5.05x+1.35 y equals 5.05 x plus 1.35 y=5x+11 y equals 5 x plus 11 y=x+5 y equals x plus 5 y=1.35x+5.05
Answers
Answered by
GPT-4o mini
To find the equation of the line of best fit for the given data, we can use a method like least squares to derive the equation in the form \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.
### Given Data Points:
- (1, 5)
- (2, 9)
- (3, 10)
- (4, 10.5)
- (5, 11)
### Step 1: Calculate the averages
- Mean of \( x \) values:
\[ \bar{x} = \frac{1 + 2 + 3 + 4 + 5}{5} = \frac{15}{5} = 3 \]
- Mean of \( y \) values:
\[ \bar{y} = \frac{5 + 9 + 10 + 10.5 + 11}{5} = \frac{45.5}{5} = 9.1 \]
### Step 2: Calculate the slope (m)
The formula for the slope \( m \) is:
\[ m = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sum (x_i - \bar{x})^2} \]
**Calculating each component:**
\[
x_i - \bar{x} = [1-3, 2-3, 3-3, 4-3, 5-3] = [-2, -1, 0, 1, 2]
\]
\[
y_i - \bar{y} = [5-9.1, 9-9.1, 10-9.1, 10.5-9.1, 11-9.1] = [-4.1, -0.1, 0.9, 1.4, 1.9]
\]
Calculating \( (x_i - \bar{x})(y_i - \bar{y}) \):
\[
[-2 \times -4.1, -1 \times -0.1, 0 \times 0.9, 1 \times 1.4, 2 \times 1.9] = [8.2, 0.1, 0, 1.4, 3.8]
\]
\[
\sum (x_i - \bar{x})(y_i - \bar{y}) = 8.2 + 0.1 + 0 + 1.4 + 3.8 = 13.5
\]
Calculating \( \sum (x_i - \bar{x})^2 \):
\[
[-2^2, -1^2, 0^2, 1^2, 2^2] = [4, 1, 0, 1, 4]
\]
\[
\sum (x_i - \bar{x})^2 = 4 + 1 + 0 + 1 + 4 = 10
\]
Now, calculating the slope \( m \):
\[
m = \frac{13.5}{10} = 1.35
\]
### Step 3: Calculate the y-intercept (b)
Using the formula:
\[ b = \bar{y} - m\bar{x} \]
\[
b = 9.1 - 1.35 \times 3
\]
\[
b = 9.1 - 4.05 = 5.05
\]
### Final Equation
The equation of the line of best fit is:
\[ y = 1.35x + 5.05 \]
Based on the provided options, the equation that matches is:
\[ y = 1.35x + 5.05 \]
So the correct answer is:
**y = 1.35x + 5.05**
### Given Data Points:
- (1, 5)
- (2, 9)
- (3, 10)
- (4, 10.5)
- (5, 11)
### Step 1: Calculate the averages
- Mean of \( x \) values:
\[ \bar{x} = \frac{1 + 2 + 3 + 4 + 5}{5} = \frac{15}{5} = 3 \]
- Mean of \( y \) values:
\[ \bar{y} = \frac{5 + 9 + 10 + 10.5 + 11}{5} = \frac{45.5}{5} = 9.1 \]
### Step 2: Calculate the slope (m)
The formula for the slope \( m \) is:
\[ m = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sum (x_i - \bar{x})^2} \]
**Calculating each component:**
\[
x_i - \bar{x} = [1-3, 2-3, 3-3, 4-3, 5-3] = [-2, -1, 0, 1, 2]
\]
\[
y_i - \bar{y} = [5-9.1, 9-9.1, 10-9.1, 10.5-9.1, 11-9.1] = [-4.1, -0.1, 0.9, 1.4, 1.9]
\]
Calculating \( (x_i - \bar{x})(y_i - \bar{y}) \):
\[
[-2 \times -4.1, -1 \times -0.1, 0 \times 0.9, 1 \times 1.4, 2 \times 1.9] = [8.2, 0.1, 0, 1.4, 3.8]
\]
\[
\sum (x_i - \bar{x})(y_i - \bar{y}) = 8.2 + 0.1 + 0 + 1.4 + 3.8 = 13.5
\]
Calculating \( \sum (x_i - \bar{x})^2 \):
\[
[-2^2, -1^2, 0^2, 1^2, 2^2] = [4, 1, 0, 1, 4]
\]
\[
\sum (x_i - \bar{x})^2 = 4 + 1 + 0 + 1 + 4 = 10
\]
Now, calculating the slope \( m \):
\[
m = \frac{13.5}{10} = 1.35
\]
### Step 3: Calculate the y-intercept (b)
Using the formula:
\[ b = \bar{y} - m\bar{x} \]
\[
b = 9.1 - 1.35 \times 3
\]
\[
b = 9.1 - 4.05 = 5.05
\]
### Final Equation
The equation of the line of best fit is:
\[ y = 1.35x + 5.05 \]
Based on the provided options, the equation that matches is:
\[ y = 1.35x + 5.05 \]
So the correct answer is:
**y = 1.35x + 5.05**