Question
Find the equation for the least squares regression line of the data described below.
Zachary wonders how many times he has played his favorite songs on his computer. To investigate, he looked up information stored by his music playing software.
Zachary determined how long he had owned each song (in months), x. He also counted how many times each song had been played, y.
Months owned Plays
6 32
10 57
12 56
17 58
26 87
28 81
Round your answers to the nearest thousandth.
y=
x+
Zachary wonders how many times he has played his favorite songs on his computer. To investigate, he looked up information stored by his music playing software.
Zachary determined how long he had owned each song (in months), x. He also counted how many times each song had been played, y.
Months owned Plays
6 32
10 57
12 56
17 58
26 87
28 81
Round your answers to the nearest thousandth.
y=
x+
Answers
GPT-4o mini
To find the equation of the least squares regression line for the given data, we need to follow these steps:
1. Calculate the means of the \( x \) values (months owned) and \( y \) values (plays).
2. Compute the slope (\( m \)) of the regression line.
3. Compute the y-intercept (\( b \)) of the regression line.
4. Formulate the equation of the least squares regression line, which is of the form \( y = mx + b \).
Given the data:
\[
\begin{align*}
\text{Months owned (x)}: & \quad 6, 10, 12, 17, 26, 28 \\
\text{Plays (y)}: & \quad 32, 57, 56, 58, 87, 81 \\
\end{align*}
\]
**Step 1: Calculate the means**
\[
\bar{x} = \frac{6 + 10 + 12 + 17 + 26 + 28}{6} = \frac{99}{6} = 16.5
\]
\[
\bar{y} = \frac{32 + 57 + 56 + 58 + 87 + 81}{6} = \frac{371}{6} \approx 61.8333
\]
**Step 2: Compute 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 \( \sum{(x_i - \bar{x})(y_i - \bar{y})} \):
\[
\begin{align*}
(6 - 16.5)(32 - 61.8333) & = (-10.5)(-29.8333) \approx 313.125 \\
(10 - 16.5)(57 - 61.8333) & = (-6.5)(-4.8333) \approx 31.375 \\
(12 - 16.5)(56 - 61.8333) & = (-4.5)(-5.8333) \approx 26.25 \\
(17 - 16.5)(58 - 61.8333) & = (0.5)(-3.8333) \approx -1.9167 \\
(26 - 16.5)(87 - 61.8333) & = (9.5)(25.1667) \approx 239.0833 \\
(28 - 16.5)(81 - 61.8333) & = (11.5)(19.1667) \approx 220.4167 \\
\end{align*}
\]
Now sum these values:
\[
\sum{(x_i - \bar{x})(y_i - \bar{y})} \approx 313.125 + 31.375 + 26.25 - 1.9167 + 239.0833 + 220.4167 \approx 828.3333
\]
Calculating \( \sum{(x_i - \bar{x})^2} \):
\[
\begin{align*}
(6 - 16.5)^2 & \approx 110.25 \\
(10 - 16.5)^2 & \approx 42.25 \\
(12 - 16.5)^2 & \approx 20.25 \\
(17 - 16.5)^2 & \approx 0.25 \\
(26 - 16.5)^2 & \approx 90.25 \\
(28 - 16.5)^2 & \approx 132.25 \\
\end{align*}
\]
Now sum these values:
\[
\sum{(x_i - \bar{x})^2} \approx 110.25 + 42.25 + 20.25 + 0.25 + 90.25 + 132.25 \approx 395.5
\]
Now, calculate the slope \( m \):
\[
m = \frac{828.3333}{395.5} \approx 2.094
\]
**Step 3: Compute the y-intercept (\( b \))**
The formula for the y-intercept \( b \) is:
\[
b = \bar{y} - m\bar{x}
\]
\[
b = 61.8333 - (2.094)(16.5) \approx 61.8333 - 34.510 \approx 27.3233
\]
**Step 4: Formulate the equation of the least squares regression line**
Now, rounding \( m \) and \( b \) to the nearest thousandth:
\[
m \approx 2.094 \quad \text{and} \quad b \approx 27.323
\]
The equation of the least squares regression line is:
\[
y = 2.094x + 27.323
\]
So the final answer is:
\[
y = 2.094x + 27.323
\]
1. Calculate the means of the \( x \) values (months owned) and \( y \) values (plays).
2. Compute the slope (\( m \)) of the regression line.
3. Compute the y-intercept (\( b \)) of the regression line.
4. Formulate the equation of the least squares regression line, which is of the form \( y = mx + b \).
Given the data:
\[
\begin{align*}
\text{Months owned (x)}: & \quad 6, 10, 12, 17, 26, 28 \\
\text{Plays (y)}: & \quad 32, 57, 56, 58, 87, 81 \\
\end{align*}
\]
**Step 1: Calculate the means**
\[
\bar{x} = \frac{6 + 10 + 12 + 17 + 26 + 28}{6} = \frac{99}{6} = 16.5
\]
\[
\bar{y} = \frac{32 + 57 + 56 + 58 + 87 + 81}{6} = \frac{371}{6} \approx 61.8333
\]
**Step 2: Compute 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 \( \sum{(x_i - \bar{x})(y_i - \bar{y})} \):
\[
\begin{align*}
(6 - 16.5)(32 - 61.8333) & = (-10.5)(-29.8333) \approx 313.125 \\
(10 - 16.5)(57 - 61.8333) & = (-6.5)(-4.8333) \approx 31.375 \\
(12 - 16.5)(56 - 61.8333) & = (-4.5)(-5.8333) \approx 26.25 \\
(17 - 16.5)(58 - 61.8333) & = (0.5)(-3.8333) \approx -1.9167 \\
(26 - 16.5)(87 - 61.8333) & = (9.5)(25.1667) \approx 239.0833 \\
(28 - 16.5)(81 - 61.8333) & = (11.5)(19.1667) \approx 220.4167 \\
\end{align*}
\]
Now sum these values:
\[
\sum{(x_i - \bar{x})(y_i - \bar{y})} \approx 313.125 + 31.375 + 26.25 - 1.9167 + 239.0833 + 220.4167 \approx 828.3333
\]
Calculating \( \sum{(x_i - \bar{x})^2} \):
\[
\begin{align*}
(6 - 16.5)^2 & \approx 110.25 \\
(10 - 16.5)^2 & \approx 42.25 \\
(12 - 16.5)^2 & \approx 20.25 \\
(17 - 16.5)^2 & \approx 0.25 \\
(26 - 16.5)^2 & \approx 90.25 \\
(28 - 16.5)^2 & \approx 132.25 \\
\end{align*}
\]
Now sum these values:
\[
\sum{(x_i - \bar{x})^2} \approx 110.25 + 42.25 + 20.25 + 0.25 + 90.25 + 132.25 \approx 395.5
\]
Now, calculate the slope \( m \):
\[
m = \frac{828.3333}{395.5} \approx 2.094
\]
**Step 3: Compute the y-intercept (\( b \))**
The formula for the y-intercept \( b \) is:
\[
b = \bar{y} - m\bar{x}
\]
\[
b = 61.8333 - (2.094)(16.5) \approx 61.8333 - 34.510 \approx 27.3233
\]
**Step 4: Formulate the equation of the least squares regression line**
Now, rounding \( m \) and \( b \) to the nearest thousandth:
\[
m \approx 2.094 \quad \text{and} \quad b \approx 27.323
\]
The equation of the least squares regression line is:
\[
y = 2.094x + 27.323
\]
So the final answer is:
\[
y = 2.094x + 27.323
\]