Question
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.
Average song length Albums sold
3.08 202,101
3.27 268,412
3.51 272,230
4.47 155,736
5.91 107,479
Round your answers to the nearest thousandth.
note round all numbers and please do not get this wrong
y= ? x+ ?
Answers
GPT-4o mini
To find the least squares regression line \(y = mx + b\), we first need to calculate the necessary summary statistics: means of \(x\) and \(y\), the slope \(m\), and the y-intercept \(b\).
1. **Calculate Means:**
- Let \(n\) be the number of data points (albums): \(n = 5\).
- Average song lengths \(x\): \(3.08, 3.27, 3.51, 4.47, 5.91\)
- Albums sold \(y\): \(202101, 268412, 272230, 155736, 107479\)
\[
\bar{x} = \frac{3.08 + 3.27 + 3.51 + 4.47 + 5.91}{5} = \frac{19.24}{5} = 3.848
\]
\[
\bar{y} = \frac{202101 + 268412 + 272230 + 155736 + 107479}{5} = \frac{1001958}{5} = 200391.6
\]
2. **Calculate the slope \(m\):**
\[
m = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sum (x_i - \bar{x})^2}
\]
- Compute \( \sum (x_i - \bar{x})(y_i - \bar{y}) \):
- \(x_1 = 3.08, y_1 = 202101\)
- \(x_2 = 3.27, y_2 = 268412\)
- \(x_3 = 3.51, y_3 = 272230\)
- \(x_4 = 4.47, y_4 = 155736\)
- \(x_5 = 5.91, y_5 = 107479\)
| \(x_i\) | \(y_i\) | \(x_i - \bar{x}\) | \(y_i - \bar{y}\) | \((x_i - \bar{x})(y_i - \bar{y})\) |
|---------|---------|-------------------|-------------------|-----------------------------------|
| 3.08 | 202101 | -0.768 | -18290.6 | 14040.8648 |
| 3.27 | 268412 | -0.578 | 68020.4 | -39250.3772 |
| 3.51 | 272230 | -0.338 | 71838.4 | -24217.2192 |
| 4.47 | 155736 | 0.622 | -44655.6 | -27759.7876 |
| 5.91 | 107479 | 2.062 | -92812.6 | -191157.2976 |
| **Total** | | | | **-183362.8354** |
\[
\sum (x_i - \bar{x})(y_i - \bar{y}) \approx -183362.84
\]
- Compute \( \sum (x_i - \bar{x})^2 \):
| \(x_i\) | \(x_i - \bar{x}\) | \((x_i - \bar{x})^2\) |
|---------|-------------------|-----------------------|
| 3.08 | -0.768 | 0.589824 |
| 3.27 | -0.578 | 0.334084 |
| 3.51 | -0.338 | 0.114244 |
| 4.47 | 0.622 | 0.387684 |
| 5.91 | 2.062 | 4.264244 |
| **Total** | | **5.68808** |
Now we can calculate the slope.
\[
m = \frac{-183362.84}{5.68808} \approx -32212.427
\]
Rounding to the nearest thousandth:
\[
m \approx -32212.427
\]
3. **Calculate the intercept \(b\):**
\[
b = \bar{y} - m\bar{x}
\]
\[
b = 200391.6 - (-32212.427)(3.848)
\]
\[
b \approx 200391.6 + 123714.397 \approx 324105.997
\]
Rounding to the nearest thousandth:
\[
b \approx 324106.000
\]
4. **Equation of the least squares regression line**:
\[
y \approx -32212.427x + 324106.000
\]
Thus, the final answer for the least squares regression line is:
\[
y = -32212.427x + 324106.000
\]
1. **Calculate Means:**
- Let \(n\) be the number of data points (albums): \(n = 5\).
- Average song lengths \(x\): \(3.08, 3.27, 3.51, 4.47, 5.91\)
- Albums sold \(y\): \(202101, 268412, 272230, 155736, 107479\)
\[
\bar{x} = \frac{3.08 + 3.27 + 3.51 + 4.47 + 5.91}{5} = \frac{19.24}{5} = 3.848
\]
\[
\bar{y} = \frac{202101 + 268412 + 272230 + 155736 + 107479}{5} = \frac{1001958}{5} = 200391.6
\]
2. **Calculate the slope \(m\):**
\[
m = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sum (x_i - \bar{x})^2}
\]
- Compute \( \sum (x_i - \bar{x})(y_i - \bar{y}) \):
- \(x_1 = 3.08, y_1 = 202101\)
- \(x_2 = 3.27, y_2 = 268412\)
- \(x_3 = 3.51, y_3 = 272230\)
- \(x_4 = 4.47, y_4 = 155736\)
- \(x_5 = 5.91, y_5 = 107479\)
| \(x_i\) | \(y_i\) | \(x_i - \bar{x}\) | \(y_i - \bar{y}\) | \((x_i - \bar{x})(y_i - \bar{y})\) |
|---------|---------|-------------------|-------------------|-----------------------------------|
| 3.08 | 202101 | -0.768 | -18290.6 | 14040.8648 |
| 3.27 | 268412 | -0.578 | 68020.4 | -39250.3772 |
| 3.51 | 272230 | -0.338 | 71838.4 | -24217.2192 |
| 4.47 | 155736 | 0.622 | -44655.6 | -27759.7876 |
| 5.91 | 107479 | 2.062 | -92812.6 | -191157.2976 |
| **Total** | | | | **-183362.8354** |
\[
\sum (x_i - \bar{x})(y_i - \bar{y}) \approx -183362.84
\]
- Compute \( \sum (x_i - \bar{x})^2 \):
| \(x_i\) | \(x_i - \bar{x}\) | \((x_i - \bar{x})^2\) |
|---------|-------------------|-----------------------|
| 3.08 | -0.768 | 0.589824 |
| 3.27 | -0.578 | 0.334084 |
| 3.51 | -0.338 | 0.114244 |
| 4.47 | 0.622 | 0.387684 |
| 5.91 | 2.062 | 4.264244 |
| **Total** | | **5.68808** |
Now we can calculate the slope.
\[
m = \frac{-183362.84}{5.68808} \approx -32212.427
\]
Rounding to the nearest thousandth:
\[
m \approx -32212.427
\]
3. **Calculate the intercept \(b\):**
\[
b = \bar{y} - m\bar{x}
\]
\[
b = 200391.6 - (-32212.427)(3.848)
\]
\[
b \approx 200391.6 + 123714.397 \approx 324105.997
\]
Rounding to the nearest thousandth:
\[
b \approx 324106.000
\]
4. **Equation of the least squares regression line**:
\[
y \approx -32212.427x + 324106.000
\]
Thus, the final answer for the least squares regression line is:
\[
y = -32212.427x + 324106.000
\]