I apologize for the confusion. Let's go through the calculation of the line of best fit step-by-step, ensuring we effectively find the correct y-intercept and double-check the slope.
Step 1: Let's summarize the data points.
We have the following data points:
- \( (225, 52.90) \)
- \( (800, 95) \)
- \( (1375, 155) \)
- \( (1675, 174) \)
- \( (1950, 204) \)
- \( (2250, 210) \)
Step 2: Calculate Mean Values
We need to find \( \bar{x} \) and \( \bar{y} \):
Calculating \( \bar{x} \):
\[
\bar{x} = \frac{225 + 800 + 1375 + 1675 + 1950 + 2250}{6} = \frac{10275}{6} = 1712.5
\]
Calculating \( \bar{y} \):
\[
\bar{y} = \frac{52.90 + 95 + 155 + 174 + 204 + 210}{6} = \frac{790.90}{6} \approx 131.82
\]
Step 3: Calculate the Slope (m)
Using the formula for slope \( m \):
\[
m = \frac{n(\sum xy) - (\sum x)(\sum y)}{n(\sum x^2) - (\sum x)^2}
\]
Where:
- \( n \) is the number of points.
- \( \sum xy \) is the sum of the products of each pair of \( x \) and \( y \).
- \( \sum x \) is the sum of the \( x \) values.
- \( \sum y \) is the sum of the \( y \) values.
- \( \sum x^2 \) is the sum of the squares of the \( x \) values.
Step 4: Calculate required sums
We will calculate \( \sum x \), \( \sum y \), \( \sum xy \), and \( \sum x^2 \):
\[
\sum x = 225 + 800 + 1375 + 1675 + 1950 + 2250 = 10275
\]
\[
\sum y = 52.90 + 95 + 155 + 174 + 204 + 210 = 790.90
\]
Now, calculate \( xy \) and \( x^2 \):
- For \( xy \):
- \( 225 \times 52.90 = 11977.5 \)
- \( 800 \times 95 = 76000 \)
- \( 1375 \times 155 = 213125 \)
- \( 1675 \times 174 = 291150 \)
- \( 1950 \times 204 = 397800 \)
- \( 2250 \times 210 = 472500 \)
\[
\sum xy = 11977.5 + 76000 + 213125 + 291150 + 397800 + 472500 = 1127552.5
\]
- For \( x^2 \):
\[
225^2 = 50625,, 800^2 = 640000,, 1375^2 = 1890625,, 1675^2 = 2805625,, 1950^2 = 3802500,, 2250^2 = 5062500
\]
\[
\sum x^2 = 50625 + 640000 + 1890625 + 2805625 + 3802500 + 5062500 = 10237275
\]
Step 5: Substitute values into the slope formula
Now, we have:
- \( n = 6 \)
- \( \sum x = 10275 \)
- \( \sum y = 790.90 \)
- \( \sum xy = 1127552.5 \)
- \( \sum x^2 = 10237275 \)
Now substituting into the formula for slope:
\[
m = \frac{6(1127552.5) - (10275)(790.9)}{6(10237275) - (10275)^2}
\]
Calculating:
\[
m = \frac{6765135 - 812057.75}{61423650 - 105575625}
\]
\[
= \frac{5953077.25}{80051750}
\]
\[
= 0.0744 ; (\text{approximately})
\]
Step 6: Calculate the y-intercept (b)
Using:
\[
b = \bar{y} - m \bar{x}
\]
\[
= 131.82 - 0.0744 \times 1712.5
\]
\[
= 131.82 - 127.49
\]
\[
\approx 4.33
\]
Final Equation of the Line of Best Fit
So, we have:
\[
y \approx 0.0744x + 4.33
\]
If rounded to two decimal places, the equation is:
\[
y = 0.07x + 4.33
\]
If you are using specific software that yields different values, please let me know. Alternatively, please share the specific coefficients you've obtained so we can identify the discrepancy. Thank you for your understanding!
5 answers