https://file.aiquickdraw.com/mathbot/file/1726168654770lfsx1lss.jpg

To find the least squares regression line for the data provided in the image, we can follow these steps:

### Step 1: Extract the Data

From the table:

- \( x \) (Average song length in minutes):
  \[
  x = [4.20, 4.42, 4.53, 4.96, 5.73, 6.93]
  \]

- \( y \) (Albums sold):
  \[
  y = [269672, 151448, 190493, 125863, 107320, 109366]
  \]

### Step 2: Apply the Formula for the Least Squares Regression Line
The equation of the regression line is:
\[
y = mx + b
\]
Where:
- \( m \) is the slope of the line
- \( b \) is the y-intercept

The formulas for \( m \) and \( b \) are:

\[
m = \frac{n(\Sigma xy) - (\Sigma x)(\Sigma y)}{n(\Sigma x^2) - (\Sigma x)^2}
\]
\[
b = \frac{(\Sigma y)(\Sigma x^2) - (\Sigma xy)(\Sigma x)}{n(\Sigma x^2) - (\Sigma x)^2}
\]

Where \( n \) is the number of data points.

### Step 3: Compute the Values
I will now calculate these values to determine the equation of the regression line.The equation of the least squares regression line for the data is:

\[
y = -43146.606x + 380297.177
\]

This means that as the average song length \(x\) increases, the number of albums sold \(y\) decreases by approximately 43,146 albums per minute of song length, with an intercept of 380,297 albums when the song length is zero.

Do you have any questions or would you like further clarification?

Here are 5 related questions:
1. How is the slope of the regression line interpreted in this context?
2. Can the regression line predict album sales accurately for future song lengths?
3. What are some limitations of using a linear regression model in this scenario?
4. How could the results be visualized to better understand the relationship?
5. What assumptions underlie the least squares regression method?

**Tip:** Always check the goodness-of-fit (e.g., \( R^2 \)) to understand how well the regression line represents the data.

Find the equation for the least squares regression line of the data described below. Members of the popular band DrumSolo are concerned that the songs on their most recent album are too long. They decide to see if there was any relationship between song length and popularity of their previous albums. For every previously released album, they measure the average song length (in minutes), x, and also note the number of albums sold, y.

Learn how to find the least squares regression line using given data on average song lengths and the number of albums sold. This example covers key statistical concepts such as linear regression, the least squares method, and interpreting results. Suitable for high school seniors or college students.

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