Sure! Let's go through the steps to identify the line of fit for the given data, and then we will interpret the results to understand what the graph should look like.
### Step 1: Calculate the Mean of [tex]\(x\)[/tex] and [tex]\(y\)[/tex]
First, we find the mean of the [tex]\(x\)[/tex] values and the mean of the [tex]\(y\)[/tex] values:
- Mean of [tex]\(x\)[/tex] ([tex]\(\bar{x}\)[/tex]):
[tex]\[
\bar{x} = 5.055555555555555
\][/tex]
- Mean of [tex]\(y\)[/tex] ([tex]\(\bar{y}\)[/tex]):
[tex]\[
\bar{y} = 5.322222222222222
\][/tex]
### Step 2: Calculate the Slope ([tex]\(m\)[/tex])
Next, the slope [tex]\(m\)[/tex] of the line of best fit is determined using the formula:
[tex]\[
m = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sum (x_i - \bar{x})^2}
\][/tex]
In this case:
- Numerator:
[tex]\[
\sum (x_i - \bar{x})(y_i - \bar{y}) = 43.288888888888884
\][/tex]
- Denominator:
[tex]\[
\sum (x_i - \bar{x})^2 = 56.72222222222223
\][/tex]
- Slope [tex]\(m\)[/tex]:
[tex]\[
m = \frac{43.288888888888884}{56.72222222222223} \approx 0.763173359451518
\][/tex]
### Step 3: Calculate the Intercept ([tex]\(b\)[/tex])
Then, we find the intercept [tex]\(b\)[/tex] using the formula:
[tex]\[
b = \bar{y} - m \cdot \bar{x}
\][/tex]
- Intercept [tex]\(b\)[/tex]:
[tex]\[
b = 5.322222222222222 - 0.763173359451518 \cdot 5.055555555555555 \approx 1.4639569049951033
\][/tex]
### Step 4: Form the Equation of the Line
The equation of the line of best fit can be written as:
[tex]\[
y = m x + b
\][/tex]
Substituting the values we obtained:
[tex]\[
y = 0.763173359451518 x + 1.4639569049951033
\][/tex]
### Step 5: Calculate the Predicted [tex]\(y\)[/tex] Values
Using the equation, we calculate the predicted [tex]\(y\)[/tex] values for each [tex]\(x\)[/tex]:
[tex]\[
\hat{y} = 0.763173359451518 x + 1.4639569049951033
\][/tex]
The predicted [tex]\(y\)[/tex] values are:
[tex]\[
\hat{y} = [2.22713026, 2.60871694, 4.13506366, 4.89823702, 5.66141038, 6.42458374, 6.42458374, 7.56934378, 7.95093046]
\][/tex]
### Conclusion
To identify a reasonable line of fit:
- The line should have a slope of approximately [tex]\(0.763173359451518\)[/tex].
- The y-intercept should be approximately [tex]\(1.4639569049951033\)[/tex].
- When plotted, the line should closely follow the trend of the data points, with the predicted values falling near the actual data points.
Given these parameters, we can now identify a graph that shows a line of fit matching these characteristics. The line in the graph should pass through the mean points of the data and lying near the actual data points as predicted above.
