Question

Find the equation for the least squares regression line of the data described below.
Colton runs a bakery whose menu lists the number of calories in each item. He wondered whether this information affects which pastries are purchased, so he collected some sales data to find out.
Colton kept track of how many calories each pastry has, x, and how many times that pastry had been purchased in the last week, y.
Calories Sales
118 129
166 296
181 228
215 499
231 410
Round your answers to the nearest thousandth.
y=
x+

Answer 1

To find the least squares regression line, we need to calculate the slope \( m \) and the y-intercept \( b \) of the line given by the equation \( y = mx + b \).

We will use the following formulas:

1. The slope \( m \):
\[
m = \frac{n(\sum xy) - (\sum x)(\sum y)}{n(\sum x^2) - (\sum x)^2}
\]

2. The y-intercept \( b \):
\[
b = \frac{\sum y - m(\sum x)}{n}
\]

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

Given your data:

| Calories (x) | Sales (y) |
|--------------|-----------|
| 118 | 129 |
| 166 | 296 |
| 181 | 228 |
| 215 | 499 |
| 231 | 410 |

Let's calculate the necessary sums.

1. **Calculate the sums**:

- \( \sum x = 118 + 166 + 181 + 215 + 231 = 1111 \)
- \( \sum y = 129 + 296 + 228 + 499 + 410 = 1562 \)
- \( \sum xy = (118 \times 129) + (166 \times 296) + (181 \times 228) + (215 \times 499) + (231 \times 410) \)

Calculating each product:
- \( 118 \times 129 = 15162 \)
- \( 166 \times 296 = 49256 \)
- \( 181 \times 228 = 41388 \)
- \( 215 \times 499 = 107485 \)
- \( 231 \times 410 = 94890 \)

Now summing these products:
- \( \sum xy = 15162 + 49256 + 41388 + 107485 + 94890 = 296191 \)

Now for \( \sum x^2 \):
- \( \sum x^2 = 118^2 + 166^2 + 181^2 + 215^2 + 231^2 \)

Calculating each square:
- \( 118^2 = 13924 \)
- \( 166^2 = 27556 \)
- \( 181^2 = 32761 \)
- \( 215^2 = 46225 \)
- \( 231^2 = 53361 \)

Now summing these squares:
- \( \sum x^2 = 13924 + 27556 + 32761 + 46225 + 53361 = 173927 \)

2. **Plugging into the formulas**:

Now we have:
- \( n = 5 \)
- \( \sum x = 1111 \)
- \( \sum y = 1562 \)
- \( \sum xy = 296191 \)
- \( \sum x^2 = 173927 \)

Now calculate \( m \):

\[
m = \frac{5(296191) - (1111)(1562)}{5(173927) - (1111)^2}
\]
\[
= \frac{1480955 - 1736762}{869635 - 1234321}
\]
\[
= \frac{-255807}{-364686}
\]

Calculating \( m \):

\[
m \approx 0.700
\]

Now calculate \( b \):
\[
b = \frac{1562 - 0.700 \times 1111}{5}
\]
\[
= \frac{1562 - 777.7}{5}
\]
\[
= \frac{784.3}{5}
\]
\[
b \approx 156.86
\]

Thus, the least squares regression line is:

\[
y = 0.700x + 156.860
\]

Rounded to the nearest thousandth, the final answer is:

\[
y = 0.700x + 156.860
\]