Question

Find the equation for the least squares regression line of the data described below. Every summer, the Lakewood Fair has a grasshopper catching contest. Grasshoppers are released into a fenced area, and participants are given five minutes to catch as many as they can. To even the playing field, the organizers developed a handicap system based on the results of this year's competition. The system takes into account the different experience levels of the contest participants. To create the system, the organizers recorded the number of times each contestant had previously participated, x, and the number of grasshoppers each contestant had caught this year, y. Previous competitions entered Grasshoppers caught 3 16 5 12 6 20 8 37 9 35 10 26 Round your answers to the nearest thousandth.

Answer 1

To find the equation of the least squares regression line, we will follow these steps:

1. **Collect the data points**:
For the grasshopper catching contest, you have the following data on the number of previous competitions participated (x) and the corresponding number of grasshoppers caught (y):

\[
\begin{align*}
(3, 16) \\
(5, 12) \\
(6, 20) \\
(8, 37) \\
(9, 35) \\
(10, 26) \\
\end{align*}
\]

2. **Calculate sums needed for regression formula**:
We need to calculate:
- \( n \) (number of points)
- \( \sum x \) (sum of x values)
- \( \sum y \) (sum of y values)
- \( \sum xy \) (sum of the product of x and y)
- \( \sum x^2 \) (sum of x squared)

Let's calculate these values:

\[
n = 6
\]

\[
\sum x = 3 + 5 + 6 + 8 + 9 + 10 = 41
\]

\[
\sum y = 16 + 12 + 20 + 37 + 35 + 26 = 146
\]

\[
\sum xy = (3 \cdot 16) + (5 \cdot 12) + (6 \cdot 20) + (8 \cdot 37) + (9 \cdot 35) + (10 \cdot 26) = 48 + 60 + 120 + 296 + 315 + 260 = 1099
\]

\[
\sum x^2 = 3^2 + 5^2 + 6^2 + 8^2 + 9^2 + 10^2 = 9 + 25 + 36 + 64 + 81 + 100 = 315
\]

3. **Plug these values into the regression formula**:
The coefficients \( a \) (slope) and \( b \) (intercept) are calculated using the formulas:

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

\[
b = \frac{(\sum y) - a(\sum x)}{n}
\]

Substituting in the calculated values:

\[
a = \frac{6(1099) - (41)(146)}{6(315) - (41)^2}
\]

\[
a = \frac{6594 - 5986}{1890 - 1681}
\]

\[
a = \frac{608}{209} \approx 2.911
\]

Now, calculating \( b \):

\[
b = \frac{146 - 2.911 \times 41}{6}
\]

\[
b = \frac{146 - 119.391}{6}
\]

\[
b = \frac{26.609}{6} \approx 4.434
\]

4. **Write the equation of the least squares regression line**:
The equation of the least squares regression line is:

\[
y = ax + b
\]

Substituting the values of \( a \) and \( b \):

\[
y \approx 2.911x + 4.434
\]

Thus, rounding to the nearest thousandth:

\[
\text{Least squares regression line equation: } y = 2.911x + 4.434
\]