Question
Find the equation for the least squares regression line of the data described below.
Vijay keeps detailed information about his model airplane collection. The data includes not only the characteristics of each model, but also the time it took to assemble it. He is thinking of buying a new model airplane and decided to look at this information to understand how long assembly might take.
From his data, Vijay found the number of pieces in each model, x, and how many minutes each took to assemble, y.
Pieces Assembly time
46 39
60 59
71 53
83 68
88 63
98 66
Round your answers to the nearest thousandth.
y=
x+
Answers
GPT-4o mini
To find the least squares regression line for the given data, we will first need to compute the necessary statistics, which include the means of \(x\) and \(y\), the sums of squares, and the slope and intercept of the regression line.
The least squares regression line is given by the equation:
\[
y = mx + b
\]
where:
- \(m\) is the slope,
- \(b\) is the y-intercept.
**Step 1: Calculate the means of \(x\) and \(y\)**
Given the data:
| Pieces (x) | Assembly time (y) |
|------------|-------------------|
| 46 | 39 |
| 60 | 59 |
| 71 | 53 |
| 83 | 68 |
| 88 | 63 |
| 98 | 66 |
Calculate the sums and means:
\[
\text{Mean of } x = \frac{46 + 60 + 71 + 83 + 88 + 98}{6} = \frac{446}{6} \approx 74.333
\]
\[
\text{Mean of } y = \frac{39 + 59 + 53 + 68 + 63 + 66}{6} = \frac{348}{6} \approx 58.000
\]
**Step 2: Calculate the slope \(m\)**
The formula for the slope \(m\) is:
\[
m = \frac{n(\sum xy) - (\sum x)(\sum y)}{n(\sum x^2) - (\sum x)^2}
\]
First, we need to compute:
- \(\sum x\)
- \(\sum y\)
- \(\sum xy\)
- \(\sum x^2\)
Calculating these:
- \(\sum x = 446\)
- \(\sum y = 348\)
- \(\sum xy = (46 \cdot 39) + (60 \cdot 59) + (71 \cdot 53) + (83 \cdot 68) + (88 \cdot 63) + (98 \cdot 66)\)
- \(= 1794 + 3540 + 3763 + 5644 + 5544 + 6468 = 26753\)
- \(\sum x^2 = 46^2 + 60^2 + 71^2 + 83^2 + 88^2 + 98^2\)
- \(= 2116 + 3600 + 5041 + 6889 + 7744 + 9604 = 35894\)
Now we substitute the values into the slope formula:
\[
n = 6
\]
\[
m = \frac{6(26753) - (446)(348)}{6(35894) - (446)^2}
\]
Calculating the numerator:
\[
\text{Numerator} = 6(26753) - (446)(348) = 160518 - 155208 = 5310
\]
Calculating the denominator:
\[
\text{Denominator} = 6(35894) - 198916 = 215364 - 198916 = 16448
\]
So,
\[
m = \frac{5310}{16448} \approx 0.323
\]
**Step 3: Calculate the intercept \(b\)**
The formula for the intercept \(b\) is:
\[
b = \text{mean of } y - m \cdot \text{mean of } x
\]
Using our means:
\[
b = 58 - (0.323)(74.333) \approx 58 - 24.007 \approx 33.993
\]
**Final Equation:**
Thus, we have:
\[
y \approx 0.323x + 33.993
\]
Rounded to the nearest thousandth, the equation of the least squares regression line is:
\[
y = 0.323x + 33.993
\]
\(\boxed{y = 0.323x + 33.993}\)
The least squares regression line is given by the equation:
\[
y = mx + b
\]
where:
- \(m\) is the slope,
- \(b\) is the y-intercept.
**Step 1: Calculate the means of \(x\) and \(y\)**
Given the data:
| Pieces (x) | Assembly time (y) |
|------------|-------------------|
| 46 | 39 |
| 60 | 59 |
| 71 | 53 |
| 83 | 68 |
| 88 | 63 |
| 98 | 66 |
Calculate the sums and means:
\[
\text{Mean of } x = \frac{46 + 60 + 71 + 83 + 88 + 98}{6} = \frac{446}{6} \approx 74.333
\]
\[
\text{Mean of } y = \frac{39 + 59 + 53 + 68 + 63 + 66}{6} = \frac{348}{6} \approx 58.000
\]
**Step 2: Calculate the slope \(m\)**
The formula for the slope \(m\) is:
\[
m = \frac{n(\sum xy) - (\sum x)(\sum y)}{n(\sum x^2) - (\sum x)^2}
\]
First, we need to compute:
- \(\sum x\)
- \(\sum y\)
- \(\sum xy\)
- \(\sum x^2\)
Calculating these:
- \(\sum x = 446\)
- \(\sum y = 348\)
- \(\sum xy = (46 \cdot 39) + (60 \cdot 59) + (71 \cdot 53) + (83 \cdot 68) + (88 \cdot 63) + (98 \cdot 66)\)
- \(= 1794 + 3540 + 3763 + 5644 + 5544 + 6468 = 26753\)
- \(\sum x^2 = 46^2 + 60^2 + 71^2 + 83^2 + 88^2 + 98^2\)
- \(= 2116 + 3600 + 5041 + 6889 + 7744 + 9604 = 35894\)
Now we substitute the values into the slope formula:
\[
n = 6
\]
\[
m = \frac{6(26753) - (446)(348)}{6(35894) - (446)^2}
\]
Calculating the numerator:
\[
\text{Numerator} = 6(26753) - (446)(348) = 160518 - 155208 = 5310
\]
Calculating the denominator:
\[
\text{Denominator} = 6(35894) - 198916 = 215364 - 198916 = 16448
\]
So,
\[
m = \frac{5310}{16448} \approx 0.323
\]
**Step 3: Calculate the intercept \(b\)**
The formula for the intercept \(b\) is:
\[
b = \text{mean of } y - m \cdot \text{mean of } x
\]
Using our means:
\[
b = 58 - (0.323)(74.333) \approx 58 - 24.007 \approx 33.993
\]
**Final Equation:**
Thus, we have:
\[
y \approx 0.323x + 33.993
\]
Rounded to the nearest thousandth, the equation of the least squares regression line is:
\[
y = 0.323x + 33.993
\]
\(\boxed{y = 0.323x + 33.993}\)