Question
Find the equation for the least squares regression line of the data described below. A software developer is looking for ways to predict how many bugs will appear in future projects. He used a bug-tracking database to analyze several recent projects. From the database, he recorded the number of lines of code for each project, x. He also looked up the number of bugs that had been found in each project's code, y. Lines of code Bugs 10,305 225 14,135 377 15,413 168 16,681 333 19,040 494 19,239 484 Round your answers to the nearest thousandth. y=x+
Answers
GPT-4o mini
To find the equation of the least squares regression line, we need to first calculate the slope (m) and intercept (b) of the line using the formula:
\[
y = mx + b
\]
Given the data points:
\[
\begin{align*}
(10305, 225) \\
(14135, 377) \\
(15413, 168) \\
(16681, 333) \\
(19040, 494) \\
(19239, 484) \\
\end{align*}
\]
Let's calculate the necessary values:
1. **Calculate the means** of \(x\) and \(y\):
\[
\bar{x} = \frac{10305 + 14135 + 15413 + 16681 + 19040 + 19239}{6} = \frac{10305 + 14135 + 15413 + 16681 + 19040 + 19239}{6} = 15132.5
\]
\[
\bar{y} = \frac{225 + 377 + 168 + 333 + 494 + 484}{6} = \frac{225 + 377 + 168 + 333 + 494 + 484}{6} = 362.8333
\]
2. **Calculate the slope \(m\)**:
The formula for slope \(m\) is:
\[
m = \frac{\sum{(x_i - \bar{x})(y_i - \bar{y})}}{\sum{(x_i - \bar{x})^2}}
\]
For each data point, we calculate \((x_i - \bar{x})\), \((y_i - \bar{y})\), and their products.
\[
\begin{align*}
(10305 - 15132.5)(225 - 362.8333) &= (-4827.5)(-137.8333) = 665658.375 \\
(14135 - 15132.5)(377 - 362.8333) &= (-997.5)(14.1667) = -14110.75 \\
(15413 - 15132.5)(168 - 362.8333) &= (280.5)(-194.8333) = -54611.67 \\
(16681 - 15132.5)(333 - 362.8333) &= (1548.5)(-29.8333) = -46186.81 \\
(19040 - 15132.5)(494 - 362.8333) &= (3917.5)(131.1667) = 513616.81 \\
(19239 - 15132.5)(484 - 362.8333) &= (4106.5)(121.1667) = 497569.825 \\
\end{align*}
\]
Summing these products:
\[
\sum{(x_i - \bar{x})(y_i - \bar{y})} = 665658.375 - 14110.75 - 54611.67 - 46186.81 + 513616.81 + 497569.825 = 882035.675
\]
Now for the squared differences:
\[
\begin{align*}
(10305 - 15132.5)^2 &= 23276156.25 \\
(14135 - 15132.5)^2 &= 996006.25 \\
(15413 - 15132.5)^2 &= 792.25 \\
(16681 - 15132.5)^2 &= 2355448.25 \\
(19040 - 15132.5)^2 &= 15376756.25 \\
(19239 - 15132.5)^2 &= 16913156.25 \\
\end{align*}
\]
Summing these squares:
\[
\sum{(x_i - \bar{x})^2} = 23276156.25 + 996006.25 + 792.25 + 2355448.25 + 15376756.25 + 16913156.25 = 43665957.5
\]
3. **Calculate the slope (m)**:
\[
m = \frac{882035.675}{43665957.5} \approx 0.0202
\]
4. **Calculate the intercept (b)**:
The formula for intercept \(b\) is:
\[
b = \bar{y} - m \bar{x}
\]
\[
b = 362.8333 - (0.0202)(15132.5) \approx 362.8333 - 306.0275 \approx 56.8058
\]
5. **Final Equation**:
Rounding to the nearest thousandth, the least squares regression line is:
\[
y = 0.020x + 56.806
\]
Therefore, the final answer is:
\[
y \approx 0.020x + 56.806
\]
\[
y = mx + b
\]
Given the data points:
\[
\begin{align*}
(10305, 225) \\
(14135, 377) \\
(15413, 168) \\
(16681, 333) \\
(19040, 494) \\
(19239, 484) \\
\end{align*}
\]
Let's calculate the necessary values:
1. **Calculate the means** of \(x\) and \(y\):
\[
\bar{x} = \frac{10305 + 14135 + 15413 + 16681 + 19040 + 19239}{6} = \frac{10305 + 14135 + 15413 + 16681 + 19040 + 19239}{6} = 15132.5
\]
\[
\bar{y} = \frac{225 + 377 + 168 + 333 + 494 + 484}{6} = \frac{225 + 377 + 168 + 333 + 494 + 484}{6} = 362.8333
\]
2. **Calculate the slope \(m\)**:
The formula for slope \(m\) is:
\[
m = \frac{\sum{(x_i - \bar{x})(y_i - \bar{y})}}{\sum{(x_i - \bar{x})^2}}
\]
For each data point, we calculate \((x_i - \bar{x})\), \((y_i - \bar{y})\), and their products.
\[
\begin{align*}
(10305 - 15132.5)(225 - 362.8333) &= (-4827.5)(-137.8333) = 665658.375 \\
(14135 - 15132.5)(377 - 362.8333) &= (-997.5)(14.1667) = -14110.75 \\
(15413 - 15132.5)(168 - 362.8333) &= (280.5)(-194.8333) = -54611.67 \\
(16681 - 15132.5)(333 - 362.8333) &= (1548.5)(-29.8333) = -46186.81 \\
(19040 - 15132.5)(494 - 362.8333) &= (3917.5)(131.1667) = 513616.81 \\
(19239 - 15132.5)(484 - 362.8333) &= (4106.5)(121.1667) = 497569.825 \\
\end{align*}
\]
Summing these products:
\[
\sum{(x_i - \bar{x})(y_i - \bar{y})} = 665658.375 - 14110.75 - 54611.67 - 46186.81 + 513616.81 + 497569.825 = 882035.675
\]
Now for the squared differences:
\[
\begin{align*}
(10305 - 15132.5)^2 &= 23276156.25 \\
(14135 - 15132.5)^2 &= 996006.25 \\
(15413 - 15132.5)^2 &= 792.25 \\
(16681 - 15132.5)^2 &= 2355448.25 \\
(19040 - 15132.5)^2 &= 15376756.25 \\
(19239 - 15132.5)^2 &= 16913156.25 \\
\end{align*}
\]
Summing these squares:
\[
\sum{(x_i - \bar{x})^2} = 23276156.25 + 996006.25 + 792.25 + 2355448.25 + 15376756.25 + 16913156.25 = 43665957.5
\]
3. **Calculate the slope (m)**:
\[
m = \frac{882035.675}{43665957.5} \approx 0.0202
\]
4. **Calculate the intercept (b)**:
The formula for intercept \(b\) is:
\[
b = \bar{y} - m \bar{x}
\]
\[
b = 362.8333 - (0.0202)(15132.5) \approx 362.8333 - 306.0275 \approx 56.8058
\]
5. **Final Equation**:
Rounding to the nearest thousandth, the least squares regression line is:
\[
y = 0.020x + 56.806
\]
Therefore, the final answer is:
\[
y \approx 0.020x + 56.806
\]