Question

Find the equation for the least squares regression line of the data described below


Dosage Rise in blood sugar level
148 29.56
320 28.45
351 27.07
364 29.60
498 17.61
796 19.46
Round your answers to the nearest thousandth.

Answer 1

To find the equation of the least squares regression line (often represented as \( y = mx + b \)), we need to compute the slope (\( m \)) and the y-intercept (\( b \)) based on the provided data points.

Here are the data points you provided:

| Dosage (x) | Rise in blood sugar level (y) |
|------------|-------------------------------|
| 148 | 29.56 |
| 320 | 28.45 |
| 351 | 27.07 |
| 364 | 29.60 |
| 498 | 17.61 |
| 796 | 19.46 |

### Step 1: Calculate Averages

1. Calculate \( n \) (the number of data points):
\[ n = 6 \]

2. Calculate the sums:
\[
\sum x = 148 + 320 + 351 + 364 + 498 + 796 = 2477
\]
\[
\sum y = 29.56 + 28.45 + 27.07 + 29.60 + 17.61 + 19.46 = 151.75
\]
\[
\sum xy = (148 \times 29.56) + (320 \times 28.45) + (351 \times 27.07) + (364 \times 29.60) + (498 \times 17.61) + (796 \times 19.46) = 148 \times 29.56 + 320 \times 28.45 + 351 \times 27.07 + 364 \times 29.60 + 498 \times 17.61 + 796 \times 19.46 = 11335.68 + 9104.00 + 9505.57 + 10784.40 + 8768.58 + 15462.16 = 63370.39
\]
\[
\sum x^2 = 148^2 + 320^2 + 351^2 + 364^2 + 498^2 + 796^2 = 21904 + 102400 + 123201 + 132496 + 248004 + 633616 = 112800
\]

### Step 2: Calculate Slope (m) and Intercept (b)

The formulas for the slope \( m \) and the y-intercept \( b \) are:

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

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

### Calculate \( m \):

Plugging the values into the formulas:

\[
m = \frac{6(63370.39) - (2477)(151.75)}{6(112800) - (2477)^2}
\]

Calculating the numerator:

\[
6(63370.39) = 380222.34
\]
\[
(2477)(151.75) = 375150.75
\]
\[
\text{Numerator} = 380222.34 - 375150.75 = 5071.59
\]

Calculating the denominator:

\[
6(112800) = 676800
\]
\[
(2477)^2 = 6131529
\]
\[
\text{Denominator} = 676800 - 6131529 = -5454729
\]

### Final Slope \( m \):

Now computing \( m \):

\[
m = \frac{5071.59}{-5454729} \approx -0.000928
\]

### Calculate \( b \):

\[
b = \frac{(151.75)(112800) - (2477)(63370.39)}{676800 - 6131529}
\]

Calculating the numerator:

\[
(151.75)(112800) = 17199900
\]
\[
(2477)(63370.39) = 157559080.83
\]
\[
\text{Numerator} = 17199900 - 157559080.83 = -140359180.83
\]

### Y-intercept \( b \):

Now computing \( b \):

\[
b = \frac{-140359180.83}{-5454729} \approx 25.691
\]

### Final Equation

The least squares regression line is:

\[
y \approx -0.000928x + 25.691
\]

### Rounding to the nearest thousandth:

Finally, rounding our answers gives:

\[
y \approx -0.001x + 25.691
\]

Thus, the least squares regression line is:

\[
\boxed{y = -0.001x + 25.691}
\]