To determine which equation best represents the line of best fit for Rob's data, we need to analyze the relationship between the font size (independent variable [tex]\( x \)[/tex]) and the word count (dependent variable [tex]\( y \)[/tex]).
Here's a detailed, step-by-step solution to identify the best fitting equation:
1. Data Overview:
- Font Size ([tex]\( x \)[/tex]): [14, 12, 16, 10, 12, 14, 16, 18, 24, 22]
- Word Count ([tex]\( y \)[/tex]): [352, 461, 340, 407, 435, 381, 280, 201, 138, 114]
2. Linear Regression Analysis:
- We perform a linear regression analysis to estimate the slope and intercept of the best-fit line, examining how well each given equation fits the data.
3. Slope, Intercept and Other Linear Regression Metrics:
- The slope ([tex]\( \text{slope} \)[/tex]) of the best-fit line is approximately [tex]\(-26.06\)[/tex].
- The intercept ([tex]\( \text{intercept} \)[/tex]) is approximately [tex]\(722.63\)[/tex].
- The correlation coefficient ([tex]\( r \)[/tex]-value) is approximately [tex]\(-0.946\)[/tex], indicating a strong negative correlation between font size and word count.
- The p-value ([tex]\( p \)[/tex]-value) is extremely small, approximately [tex]\(3.51 \times 10^{-5}\)[/tex], indicating that the slope is statistically significant.
- The standard error of the slope is approximately [tex]\(3.16\)[/tex].
4. Comparing the Given Equations:
- We compare the given equations:
1. [tex]\( y = -55x + 407 \)[/tex]
2. [tex]\( y = -41x + 814 \)[/tex]
3. [tex]\( y = -38x + 922 \)[/tex]
4. [tex]\( y = -26x + 723 \)[/tex]
- We calculate the Residual Sum of Squares (RSS) for each equation to see which one fits the data best.
- RSS measures how much the observed values deviate from the values predicted by the model; lower RSS indicates a better fit.
5. Residual Sum of Squares (RSS) for Each Equation:
- The RSS for each equation are as follows:
- [tex]\( y = -55x + 407 \)[/tex] results in RSS [tex]\(6,138,525\)[/tex]
- [tex]\( y = -41x + 814 \)[/tex] results in RSS [tex]\(263,825\)[/tex]
- [tex]\( y = -38x + 922 \)[/tex] results in RSS [tex]\(41,105\)[/tex]
- [tex]\( y = -26x + 723 \)[/tex] results in RSS [tex]\(14,369\)[/tex]
6. Determining the Best Fit:
- Among the given equations, the one with the smallest RSS is the best fit.
- The equation [tex]\(y = -26x + 723\)[/tex] has the smallest RSS ([tex]\(14,369\)[/tex]), making it the best representation of the linear relationship between font size and word count based on the given data.
Therefore, the equation that represents the approximate line of best fit for Rob's data is:
[tex]\[ y = -26x + 723 \][/tex]
