Question

Find the equation for the least squares regression line of the data described below.
Grape growers in France think they will have a massive crop this year and want to know how this will influence demand for grape juice.
They compared how many grapes were produced (in millions of metric tons), x, and the amount of French grape juice that was sold in previous years (in billions of liters), y.
Grape production Grape juice sold
3.51 5.37
4.29 5.38
4.58 5.80
4.89 5.41
5.76 5.99
Round your answers to the nearest thousandth.
y=
x+

Answer 1

To find the equation for the least squares regression line, we first need to calculate the necessary statistics (means, variances, and covariances) from the provided data:

| Grape Production (x) | Grape Juice Sold (y) |
|----------------------|----------------------|
| 3.51 | 5.37 |
| 4.29 | 5.38 |
| 4.58 | 5.80 |
| 4.89 | 5.41 |
| 5.76 | 5.99 |

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

\[
\text{Mean of } x (\bar{x}) = \frac{3.51 + 4.29 + 4.58 + 4.89 + 5.76}{5} = \frac{23.03}{5} = 4.606
\]
\[
\text{Mean of } y (\bar{y}) = \frac{5.37 + 5.38 + 5.80 + 5.41 + 5.99}{5} = \frac{27.35}{5} = 5.47
\]

### Step 2: Calculate the sums needed for the slope (m)

- \( \sum x^2 \) and \( \sum y^2 \)
- \( \sum xy \)

Calculating these values,

\[
\sum x = 3.51 + 4.29 + 4.58 + 4.89 + 5.76 = 23.03
\]
\[
\sum y = 5.37 + 5.38 + 5.80 + 5.41 + 5.99 = 27.35
\]
\[
\sum x^2 = (3.51^2 + 4.29^2 + 4.58^2 + 4.89^2 + 5.76^2) = 12.3201 + 18.4041 + 20.9764 + 23.9121 + 33.0576 = 108.6703
\]
\[
\sum y^2 = (5.37^2 + 5.38^2 + 5.80^2 + 5.41^2 + 5.99^2) = 28.8369 + 28.9444 + 33.64 + 29.2681 + 35.8801 = 156.5695
\]
\[
\sum xy = (3.51 \cdot 5.37 + 4.29 \cdot 5.38 + 4.58 \cdot 5.80 + 4.89 \cdot 5.41 + 5.76 \cdot 5.99) = 18.8917 + 23.0479 + 26.5536 + 26.4939 + 34.5036 = 129.4867
\]

### Step 3: Calculate the slope \( m \)

Using the formula for the slope \( m \):

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

where \( n = 5 \):

\[
m = \frac{5(129.4867) - (23.03)(27.35)}{5(108.6703) - (23.03)^2}
\]

Calculating the numerator:

\[
5 \cdot 129.4867 = 647.4335
\]
\[
23.03 \cdot 27.35 = 628.8855
\]
\[
\text{Numerator} = 647.4335 - 628.8855 = 18.548
\]

Calculating the denominator:

\[
5 \cdot 108.6703 = 543.3515
\]
\[
(23.03)^2 = 529.4209
\]
\[
\text{Denominator} = 543.3515 - 529.4209 = 13.9306
\]

So,

\[
m = \frac{18.548}{13.9306} \approx 1.332
\]

### Step 4: Calculate y-intercept \( b \)

Using the formula \( b = \bar{y} - m \bar{x} \):

\[
b = 5.47 - 1.332 \cdot 4.606 \approx 5.47 - 6.1366 \approx -0.666
\]

### Final Result

The least squares regression line is given by:

\[
y = mx + b
\]
\[
y = 1.332x - 0.666
\]

Rounding to the nearest thousandth:

\[
y = 1.332x - 0.666
\]