Question

Find the equation for the least squares regression line of the data described below.
A pharmaceutical company is testing a new headache medication. Researchers at the company conducted an observational study using volunteer participants. All participants were given a supply of tablets of the new medication.
The participants then recorded how many headaches they had had for a week, x, and how many tablets they had taken, y.
Headaches Tablets
2 4
3 5
5 3
6 6
6 6
10 7
Round your answers to the nearest thousandth.
y=
x+

Answer 1

To find the least squares regression line (LSRL) for the given data points, we typically follow these steps:

1. **Collect the data points**: We have the following pairs (x, y):
- (4, 2)
- (5, 3)
- (3, 5)
- (6, 6)
- (6, 6)
- (7, 10)

2. **Calculate the necessary sums**:
- \( n = 6 \) (the number of data points)
- \( S_{x} = 4 + 5 + 3 + 6 + 6 + 7 = 31 \)
- \( S_{y} = 2 + 3 + 5 + 6 + 6 + 10 = 32 \)
- \( S_{xy} = 4 \cdot 2 + 5 \cdot 3 + 3 \cdot 5 + 6 \cdot 6 + 6 \cdot 6 + 7 \cdot 10 = 8 + 15 + 15 + 36 + 36 + 70 = 180 \)
- \( S_{x^2} = 4^2 + 5^2 + 3^2 + 6^2 + 6^2 + 7^2 = 16 + 25 + 9 + 36 + 36 + 49 = 171 \)
- \( S_{y^2} = 2^2 + 3^2 + 5^2 + 6^2 + 6^2 + 10^2 = 4 + 9 + 25 + 36 + 36 + 100 = 210 \)

3. **Calculate the slope (b) and intercept (a)** using the formulas:
\[
b = \frac{n(S_{xy}) - S_{x}S_{y}}{n(S_{x^2}) - (S_{x})^2}
\]
\[
a = \frac{S_{y} - bS_{x}}{n}
\]

Plugging in the numbers:
\[
b = \frac{6(180) - (31)(32)}{6(171) - (31)^2}
\]
\[
= \frac{1080 - 992}{1026 - 961}
\]
\[
= \frac{88}{65} \approx 1.354
\]

Now calculate \( a \):
\[
a = \frac{32 - (1.354)(31)}{6}
\]
\[
= \frac{32 - 41.974}{6} \approx \frac{-9.974}{6} \approx -1.662
\]

4. **Final regression equation**:
Given \( y = bx + a \), substituting the coefficients,
\[
y \approx 1.354x - 1.662
\]

Thus, rounded to the nearest thousandth, the least squares regression line is expressed as:

\[
y \approx 1.354x - 1.662
\]