Question

Find the equation for the least squares regression line of the data described below.
The owner of a bicycle store wonders if the employees who take the longest lunch breaks also make the fewest bicycle sales.
One day, he notes the length of each salesperson's lunch break (in minutes), x, as well as the number of bicycles he or she sold that day, y.
Length of lunch break (in minutes) Bicycles
24 6
25 5
29 8
35 3
40 2
41 2
Round your answers to the nearest thousandth.
y=
x+

Answer 1

To find the least squares regression line, we need to determine the slope (\(m\)) and the y-intercept (\(b\)) of the line represented as:

\[
y = mx + b
\]

### Step 1: Calculate means

First, we need to find the averages of \(x\) (length of lunch break) and \(y\) (bicycles sold).

- Given data points:
- (24, 6)
- (25, 5)
- (29, 8)
- (35, 3)
- (40, 2)
- (41, 2)

Calculating \(\bar{x}\) and \(\bar{y}\):

\[
\bar{x} = \frac{24 + 25 + 29 + 35 + 40 + 41}{6} = \frac{224}{6} = 37.333
\]

\[
\bar{y} = \frac{6 + 5 + 8 + 3 + 2 + 2}{6} = \frac{26}{6} = 4.333
\]

### Step 2: Calculate slope (m)

The formula for the slope \(m\) is:

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

Where \(N\) is the number of data points.

1. Calculate \( \sum x \), \( \sum y \), \( \sum xy \), and \( \sum x^2 \):

\[
\sum x = 24 + 25 + 29 + 35 + 40 + 41 = 224
\]

\[
\sum y = 6 + 5 + 8 + 3 + 2 + 2 = 26
\]

\[
\sum xy = (24)(6) + (25)(5) + (29)(8) + (35)(3) + (40)(2) + (41)(2) = 144 + 125 + 232 + 105 + 80 + 82 = 768
\]

\[
\sum x^2 = 24^2 + 25^2 + 29^2 + 35^2 + 40^2 + 41^2 = 576 + 625 + 841 + 1225 + 1600 + 1681 = 5,508
\]

Thus, substituting back into the slope formula:

\[
m = \frac{6(768) - (224)(26)}{6(5508) - (224)^2}
\]

Calculating the components:

\[
6(768) = 4608
\]
\[
224(26) = 5824
\]
\[
6(5508) = 33048
\]
\[
(224)^2 = 50176
\]

Thus,

\[
m = \frac{4608 - 5824}{33048 - 50176} = \frac{-1216}{-17128} \approx 0.0711
\]

### Step 3: Calculate y-intercept (b)

The formula for the y-intercept is:

\[
b = \bar{y} - m \bar{x}
\]

Calculating:

\[
b = 4.333 - 0.0711 \times 37.333
\]

Calculating \(0.0711 \times 37.333 \approx 2.6502\):

\[
b \approx 4.333 - 2.6502 \approx 1.6828
\]

### Final Equation

Thus, rounding to the nearest thousandths, the equation of the least squares regression line becomes:

\[
y = 0.071x + 1.683
\]

So, the final answer is:

\[
\boxed{y = 0.071x + 1.683}
\]