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
3.4 5.3
3.8 5.4
4.2 5.3
5.5 5.8
5.9 5.9
Round your answers to the nearest thousandth.
y=
x+
Answers
GPT-4o mini
To find the least squares regression line of the given data, we need to calculate the slope (m) and the y-intercept (b) of the line. The formula for the least squares regression line is:
\[
y = mx + b
\]
### Given Data:
- \( (x_1, y_1) = (3.4, 5.3) \)
- \( (x_2, y_2) = (3.8, 5.4) \)
- \( (x_3, y_3) = (4.2, 5.3) \)
- \( (x_4, y_4) = (5.5, 5.8) \)
- \( (x_5, y_5) = (5.9, 5.9) \)
### Step 1: Calculate the means of x and y
\[
\bar{x} = \frac{3.4 + 3.8 + 4.2 + 5.5 + 5.9}{5} = \frac{23.8}{5} = 4.76
\]
\[
\bar{y} = \frac{5.3 + 5.4 + 5.3 + 5.8 + 5.9}{5} = \frac{27.7}{5} = 5.54
\]
### Step 2: Calculate the slope (m)
The slope \( m \) is given by the formula:
\[
m = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sum (x_i - \bar{x})^2}
\]
#### Calculate the terms:
1. \( (x_i - \bar{x})(y_i - \bar{y}) \) calculation:
\[
\begin{align*}
(3.4 - 4.76)(5.3 - 5.54) & = (-1.36)(-0.24) = 0.3264 \\
(3.8 - 4.76)(5.4 - 5.54) & = (-0.96)(-0.14) = 0.1344 \\
(4.2 - 4.76)(5.3 - 5.54) & = (-0.56)(-0.24) = 0.1344 \\
(5.5 - 4.76)(5.8 - 5.54) & = (0.74)(0.26) = 0.1924 \\
(5.9 - 4.76)(5.9 - 5.54) & = (1.14)(0.36) = 0.4104 \\
\end{align*}
\]
Sum of \( (x_i - \bar{x})(y_i - \bar{y}) \):
\[
0.3264 + 0.1344 + 0.1344 + 0.1924 + 0.4104 = 1.198
\]
2. \( (x_i - \bar{x})^2 \) calculation:
\[
\begin{align*}
(3.4 - 4.76)^2 & = (-1.36)^2 = 1.8496 \\
(3.8 - 4.76)^2 & = (-0.96)^2 = 0.9216 \\
(4.2 - 4.76)^2 & = (-0.56)^2 = 0.3136 \\
(5.5 - 4.76)^2 & = (0.74)^2 = 0.5476 \\
(5.9 - 4.76)^2 & = (1.14)^2 = 1.2996 \\
\end{align*}
\]
Sum of \( (x_i - \bar{x})^2 \):
\[
1.8496 + 0.9216 + 0.3136 + 0.5476 + 1.2996 = 4.932
\]
#### Calculate \( m \):
\[
m = \frac{1.198}{4.932} \approx 0.243
\]
### Step 3: Calculate the y-intercept (b)
The y-intercept \( b \) is given by the formula:
\[
b = \bar{y} - m\bar{x}
\]
\[
b = 5.54 - (0.243)(4.76) \approx 5.54 - 1.15748 \approx 4.38252
\]
### Final Regression Equation
Rounding \( m \) and \( b \) to the nearest thousandth:
\[
m \approx 0.243, \quad b \approx 4.383
\]
Thus, the least squares regression line is:
\[
y = 0.243x + 4.383
\]
\[
y = mx + b
\]
### Given Data:
- \( (x_1, y_1) = (3.4, 5.3) \)
- \( (x_2, y_2) = (3.8, 5.4) \)
- \( (x_3, y_3) = (4.2, 5.3) \)
- \( (x_4, y_4) = (5.5, 5.8) \)
- \( (x_5, y_5) = (5.9, 5.9) \)
### Step 1: Calculate the means of x and y
\[
\bar{x} = \frac{3.4 + 3.8 + 4.2 + 5.5 + 5.9}{5} = \frac{23.8}{5} = 4.76
\]
\[
\bar{y} = \frac{5.3 + 5.4 + 5.3 + 5.8 + 5.9}{5} = \frac{27.7}{5} = 5.54
\]
### Step 2: Calculate the slope (m)
The slope \( m \) is given by the formula:
\[
m = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sum (x_i - \bar{x})^2}
\]
#### Calculate the terms:
1. \( (x_i - \bar{x})(y_i - \bar{y}) \) calculation:
\[
\begin{align*}
(3.4 - 4.76)(5.3 - 5.54) & = (-1.36)(-0.24) = 0.3264 \\
(3.8 - 4.76)(5.4 - 5.54) & = (-0.96)(-0.14) = 0.1344 \\
(4.2 - 4.76)(5.3 - 5.54) & = (-0.56)(-0.24) = 0.1344 \\
(5.5 - 4.76)(5.8 - 5.54) & = (0.74)(0.26) = 0.1924 \\
(5.9 - 4.76)(5.9 - 5.54) & = (1.14)(0.36) = 0.4104 \\
\end{align*}
\]
Sum of \( (x_i - \bar{x})(y_i - \bar{y}) \):
\[
0.3264 + 0.1344 + 0.1344 + 0.1924 + 0.4104 = 1.198
\]
2. \( (x_i - \bar{x})^2 \) calculation:
\[
\begin{align*}
(3.4 - 4.76)^2 & = (-1.36)^2 = 1.8496 \\
(3.8 - 4.76)^2 & = (-0.96)^2 = 0.9216 \\
(4.2 - 4.76)^2 & = (-0.56)^2 = 0.3136 \\
(5.5 - 4.76)^2 & = (0.74)^2 = 0.5476 \\
(5.9 - 4.76)^2 & = (1.14)^2 = 1.2996 \\
\end{align*}
\]
Sum of \( (x_i - \bar{x})^2 \):
\[
1.8496 + 0.9216 + 0.3136 + 0.5476 + 1.2996 = 4.932
\]
#### Calculate \( m \):
\[
m = \frac{1.198}{4.932} \approx 0.243
\]
### Step 3: Calculate the y-intercept (b)
The y-intercept \( b \) is given by the formula:
\[
b = \bar{y} - m\bar{x}
\]
\[
b = 5.54 - (0.243)(4.76) \approx 5.54 - 1.15748 \approx 4.38252
\]
### Final Regression Equation
Rounding \( m \) and \( b \) to the nearest thousandth:
\[
m \approx 0.243, \quad b \approx 4.383
\]
Thus, the least squares regression line is:
\[
y = 0.243x + 4.383
\]