Question

Find the equation for the least squares regression line of the data described below.
Raymond and Wanda are judges for the Greenpoint Ice Staking Federation. Due to claims of malfeasance at recent competitions, a reporter is investigating the relationship between the scores awarded by the two judges.
She has collected the scores awarded by Raymond, x, and Wanda, y, for each performance in the last competition.
Raymond's score Wanda's score
0.45 2.55
1.10 1.71
1.83 3.02
3.15 2.90
3.58 2.80
4.42 3.67
Round your answers to the nearest thousandth.
y=
x+

Answer 1

To find the least squares regression line of the given data, we can follow these steps:

1. Calculate the means of \(x\) and \(y\).
2. Calculate the slope \(m\) of the regression line.
3. Calculate the y-intercept \(b\).
4. Write the equation of the least squares regression line in the form \(y = mx + b\).

### Step 1: Calculate the Means

Given data points:

- Raymond's scores (\(x\)):
- 0.45, 1.10, 1.83, 3.15, 3.58, 4.42

- Wanda's scores (\(y\)):
- 2.55, 1.71, 3.02, 2.90, 2.80, 3.67

Calculating the means:

\[
\text{Mean of } x = \frac{0.45 + 1.10 + 1.83 + 3.15 + 3.58 + 4.42}{6} = \frac{14.53}{6} \approx 2.422
\]

\[
\text{Mean of } y = \frac{2.55 + 1.71 + 3.02 + 2.90 + 2.80 + 3.67}{6} = \frac{16.75}{6} \approx 2.792
\]

### Step 2: Calculate the Slope \(m\)

The formula for the slope \(m\) is given by:

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

Where:
- \(N\) is the number of data points.
- \(\sum xy\) is the sum of the product of each pair \((x_i \cdot y_i)\).
- \(\sum x\) is the sum of all \(x\) values.
- \(\sum y\) is the sum of all \(y\) values.
- \(\sum x^2\) is the sum of squares of \(x\).

First, we calculate the necessary sums:

\[
\sum x = 0.45 + 1.10 + 1.83 + 3.15 + 3.58 + 4.42 = 14.53
\]
\[
\sum y = 2.55 + 1.71 + 3.02 + 2.90 + 2.80 + 3.67 = 16.75
\]
\[
\sum xy = (0.45 \cdot 2.55) + (1.10 \cdot 1.71) + (1.83 \cdot 3.02) + (3.15 \cdot 2.90) + (3.58 \cdot 2.80) + (4.42 \cdot 3.67)
\]
\[
= 1.1475 + 1.881 + 5.5266 + 9.135 + 10.024 + 16.2054 = 43.9155
\]
\[
\sum x^2 = (0.45^2) + (1.10^2) + (1.83^2) + (3.15^2) + (3.58^2) + (4.42^2)
\]
\[
= 0.2025 + 1.21 + 3.3489 + 9.9225 + 12.8164 + 19.5364 = 46.0313
\]

Substituting these values into the slope formula:

\[
m = \frac{6(43.9155) - (14.53)(16.75)}{6(46.0313) - (14.53)^2}
\]

Calculating the parts step-by-step:

\[
6(43.9155) = 263.493
\]
\[
(14.53)(16.75) = 243.4275
\]
\[
6(46.0313) = 276.1878
\]
\[
(14.53)^2 = 211.3209
\]

Now calculating \(m\):

\[
m = \frac{263.493 - 243.4275}{276.1878 - 211.3209}
\]
\[
m = \frac{20.0655}{64.8669} \approx 0.309
\]

### Step 3: Calculate the y-intercept \(b\)

The formula for the y-intercept \(b\) is:

\[
b = \bar{y} - m \bar{x}
\]

Substituting the values:

\[
b = 2.792 - (0.309)(2.422)
\]
\[
b = 2.792 - 0.747918 \approx 2.044
\]

### Step 4: Write the Equation

Now we can put it all together:

\[
y = 0.309x + 2.044
\]

Rounding to the nearest thousandth:
\[
y = 0.309x + 2.044
\]

Thus, the answer is:
\[
y = 0.309x + 2.044
\]