Question

Find the equation for the least squares regression line of the data described below.
A television sports commentator recently declared that basketball teams that rely on the three-point shot cannot win championships. To see if the commentator had a valid point, a sports analyst reviewed game statistics from several teams over the past season.
For each of the teams, the analyst calculated what percentage of attempted field goals had been three-pointers, x, and the number of wins in the season, y.
Percentage of shots taken from three-point range Number of wins in a season
23 34
24 25
26 31
27 46
28 46
Round your answers to the nearest thousandth.
y=
x+

Answer 1

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

1. **Calculate the necessary sums and averages**.
2. **Calculate the slope (m) and y-intercept (b)** for the regression line using the formulas:
- Slope (m):
\[
m = \frac{n(\sum xy) - (\sum x)(\sum y)}{n(\sum x^2) - (\sum x)^2}
\]
- Intercept (b):
\[
b = \frac{\sum y - m(\sum x)}{n}
\]

3. **Formulate the equation of the regression line**:
\[
y = mx + b
\]

### Given data:
- \( (x_1, y_1) = (23, 34) \)
- \( (x_2, y_2) = (24, 25) \)
- \( (x_3, y_3) = (26, 31) \)
- \( (x_4, y_4) = (27, 46) \)
- \( (x_5, y_5) = (28, 46) \)

### Step 1: Calculate the necessary sums

- \( n = 5 \) (number of data points)
- \(\sum x = 23 + 24 + 26 + 27 + 28 = 128\)
- \(\sum y = 34 + 25 + 31 + 46 + 46 = 182\)
- \(\sum xy = 23 \times 34 + 24 \times 25 + 26 \times 31 + 27 \times 46 + 28 \times 46\)
- \(= 782 + 600 + 806 + 1242 + 1288 = 3718\)
- \(\sum x^2 = 23^2 + 24^2 + 26^2 + 27^2 + 28^2\)
- \(= 529 + 576 + 676 + 729 + 784 = 3294\)

### Step 2: Calculate slope (m) and intercept (b)

Plugging these values into the formulas:

1. Calculate the slope \( m \):
\[
m = \frac{5(3718) - (128)(182)}{5(3294) - (128)^2}
\]

2. Calculate numerators and denominators:
- \( 5 \times 3718 = 18590 \)
- \( 128 \times 182 = 23336 \)
- \( 5 \times 3294 = 16470 \)
- \( 128^2 = 16384 \)

So,
\[
m = \frac{18590 - 23336}{16470 - 16384} = \frac{-4736}{86} \approx -55.3
\]

3. Calculate intercept \( b \):
\[
b = \frac{182 - (-55.3)(128)}{5} = \frac{182 + 7078.4}{5} = \frac{7260.4}{5} \approx 1452.08
\]

### Step 3: Formulate the equation of the regression line

Thus, the least squares regression line is:
\[
y = -55.3x + 1452.08
\]

### Final Answer
Rounding to the nearest thousandth:
\[
y = -55.305x + 1452.080
\]