Question

Find the equation for the least squares regression line of the data described below.
Becky is a PR agent for an up-and-coming band. She wants to convince the band members that being active on social media is good for their careers. She claims that a large social media following typically translates into ticket sales. To prove her point, Becky looked up information about several successful bands.
She compared the number of social media followers (in millions), x, to the average number of hours it takes these bands to sell out a concert, y.
Followers (in millions) Hours
1.7 87
3.8 19
7.3 18
9.0 8
9.4 17
Round your answers to the nearest thousandth.
y=
x+

Answer 1

To find the equation of the least squares regression line \(y = mx + b\) for the given data, we first need to organize the data into two lists: the number of social media followers \(x\) and the corresponding hours to sell out concert \(y\).

Here we have the data:

| Followers (Millions) | Hours |
|---------------------|-------|
| 1.7 | 87 |
| 3.8 | 19 |
| 7.3 | 18 |
| 9.0 | 8 |
| 9.4 | 17 |

### Step 1: Calculate Necessary Sums

1. **Calculate the sums**:

- \(n = 5\) (number of data points)
- \(\sum x = 1.7 + 3.8 + 7.3 + 9.0 + 9.4 = 31.2\)
- \(\sum y = 87 + 19 + 18 + 8 + 17 = 149\)
- \(\sum xy = (1.7 \cdot 87) + (3.8 \cdot 19) + (7.3 \cdot 18) + (9.0 \cdot 8) + (9.4 \cdot 17)\)
- \(\sum x^2 = (1.7^2) + (3.8^2) + (7.3^2) + (9.0^2) + (9.4^2)\)

### Step 2: Calculate Each of these Values

1. **Calculate \(\sum xy\)**:

\[
\sum xy = (1.7 \cdot 87) + (3.8 \cdot 19) + (7.3 \cdot 18) + (9.0 \cdot 8) + (9.4 \cdot 17)
\]
\[
= 147.9 + 72.2 + 131.4 + 72.0 + 159.8 = 583.3
\]

2. **Calculate \(\sum x^2\)**:

\[
\sum x^2 = (1.7^2) + (3.8^2) + (7.3^2) + (9.0^2) + (9.4^2)
\]
\[
= 2.89 + 14.44 + 53.29 + 81.00 + 88.36 = 140.98
\]

### Step 3: Calculate the Slope \(m\) and Intercept \(b\)

- The formulas for the coefficients in least squares regression are:

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

Now substituting in:

\[
m = \frac{5(583.3) - (31.2)(149)}{5(140.98) - (31.2)^2}
\]
Calculating the values:

- \(n(\sum xy) = 5 \cdot 583.3 = 2916.5\)
- \((\sum x)(\sum y) = 31.2 \cdot 149 = 4648.8\)
- \(5(140.98) = 704.9\)
- \((\sum x)^2 = (31.2)^2 = 973.44\)

So we have:

\[
m = \frac{2916.5 - 4648.8}{704.9 - 973.44} = \frac{-1732.3}{-268.54} \approx 6.45
\]

Now to find \(b\):

\[
b = \frac{149 - 6.45(31.2)}{5}
\]
Calculating \(6.45(31.2) \approx 201.84\):

\[
b = \frac{149 - 201.84}{5} = \frac{-52.84}{5} \approx -10.568
\]

### Final Equation

Putting it all together, the least squares regression line is:

\[
y = 6.450x - 10.568
\]

Rounding to the nearest thousandth, the final equation is:

\[
y = 6.450x - 10.568
\]