Question

Find the equation for the least squares regression line of the data described below.
The CEO of McLean's Natural Fruit Juice wants to see whether the company's internal tasters give consistent ratings. She set up a blind taste test in which the tasters rated several samples of fruit juice in the same order, not knowing that two of the samples were actually the same pomegranate-kiwi blend.
Each taster rated the samples on a 100-point scale. The CEO recorded the ratings that had been given to the first sample of the pomegranate-kiwi blend, x, and the second, y.
First rating Second rating
81 82
82 85
82 79
93 85
94 87
Round your answers to the nearest thousandth.
y=
x+

Answer 1

To find the least squares regression line, we first need to calculate the means and sums necessary to find the slope (m) and intercept (b) of the line \(y = mx + b\).

Given data points:

| x (First rating) | y (Second rating) |
|------------------|--------------------|
| 81 | 82 |
| 82 | 85 |
| 82 | 79 |
| 93 | 85 |
| 94 | 87 |

### Step 1: Calculate the means of x and y

\[
\text{Mean of } x (\bar{x}) = \frac{81 + 82 + 82 + 93 + 94}{5} = \frac{432}{5} = 86.4
\]

\[
\text{Mean of } y (\bar{y}) = \frac{82 + 85 + 79 + 85 + 87}{5} = \frac{418}{5} = 83.6
\]

### Step 2: Calculate the slope (m)

Using the formula for the slope:
\[
m = \frac{n(\sum xy) - (\sum x)(\sum y)}{n(\sum x^2) - (\sum x)^2}
\]

Where:
- \(n\) is the number of data points
- \(\sum xy\) is the sum of the product of \(x\) and \(y\)
- \(\sum x\) is the sum of \(x\)
- \(\sum y\) is the sum of \(y\)
- \(\sum x^2\) is the sum of squares of \(x\)

Calculating the required sums:
\[
\sum x = 81 + 82 + 82 + 93 + 94 = 432
\]
\[
\sum y = 82 + 85 + 79 + 85 + 87 = 418
\]
\[
\sum xy = (81 \cdot 82) + (82 \cdot 85) + (82 \cdot 79) + (93 \cdot 85) + (94 \cdot 87) = 6642 + 6970 + 6478 + 7905 + 8178 = 38973
\]
\[
\sum x^2 = 81^2 + 82^2 + 82^2 + 93^2 + 94^2 = 6561 + 6724 + 6724 + 8649 + 8836 = 38494
\]

Now, plug these values into the slope formula:
\[
n = 5
\]
\[
m = \frac{5(38973) - (432)(418)}{5(38494) - (432)^2} = \frac{194865 - 180576}{192470 - 186624} = \frac{12989}{6846} \approx 1.897
\]

### Step 3: Calculate the intercept (b)

Using the formula:
\[
b = \bar{y} - m\bar{x}
\]
\[
b = 83.6 - (1.897)(86.4) \approx 83.6 - 164.1048 \approx -80.5048
\]

### Final Equation

Rounding to the nearest thousandth, we find:
- Slope (m): \(1.897\)
- Intercept (b): \(-80.505\)

Thus, the equation for the least squares regression line is:
\[
y = 1.897x - 80.505
\]