Question
Find the equation for the least squares regression line of the data described below.
A reporter for a college magazine is writing a feature about student participation in on-campus organizations. He wants to see if there is a relationship between student population and the number of on-campus organizations.
To begin, he recorded the number of students, x, and the number of officially recognized student organizations, y, for several colleges.
Students Student organizations
3,098 342
3,502 204
3,666 317
5,723 377
6,583 456
7,081 478
8,356 495
9,776 498
Round your answers to the nearest thousandth.
y=
x+
Answers
GPT-4o mini
To find the least squares regression line for the provided data, we need to use the least squares formula to derive the slope \( m \) and the y-intercept \( b \).
### Step 1: Prepare the Data
We are given the following pairs of data (students, organizations):
- (3098, 342)
- (3502, 204)
- (3666, 317)
- (5723, 377)
- (6583, 456)
- (7081, 478)
- (8356, 495)
- (9776, 498)
Let's denote our data:
\[
\text{Students } (x): [3098, 3502, 3666, 5723, 6583, 7081, 8356, 9776]
\]
\[
\text{Organizations } (y): [342, 204, 317, 377, 456, 478, 495, 498]
\]
### Step 2: Calculate Necessary Sums for Least Squares
We need the following sums for calculating the slope and intercept:
1. \( n = \) the number of data points
2. \( \sum x \) (sum of all x values)
3. \( \sum y \) (sum of all y values)
4. \( \sum xy \) (sum of the product of x and y)
5. \( \sum x^2 \) (sum of squares of x values)
Let's calculate these:
- \( n = 8 \)
- \( \sum x = 3098 + 3502 + 3666 + 5723 + 6583 + 7081 + 8356 + 9776 = 58885 \)
- \( \sum y = 342 + 204 + 317 + 377 + 456 + 478 + 495 + 498 = 3177 \)
- \( \sum xy = (3098 \cdot 342) + (3502 \cdot 204) + (3666 \cdot 317) + (5723 \cdot 377) + (6583 \cdot 456) + (7081 \cdot 478) + (8356 \cdot 495) + (9776 \cdot 498) = 11302760 \)
- \( \sum x^2 = (3098^2) + (3502^2) + (3666^2) + (5723^2) + (6583^2) + (7081^2) + (8356^2) + (9776^2) = 415845139 \)
### Step 3: Calculate the Slope (\( m \)) and Intercept (\( b \))
Using the formulas for the slope \( m \) and intercept \( b \):
\[
m = \frac{n(\sum xy) - (\sum x)(\sum y)}{n(\sum x^2) - (\sum x)^2}
\]
\[
b = \frac{\sum y - m(\sum x)}{n}
\]
Plugging in the values:
1. **Calculate the slope**:
\[
m = \frac{8(11302760) - (58885)(3177)}{8(415845139) - (58885)^2}
\]
Calculating step-by-step:
- \( 8(11302760) = 90422080 \)
- \( (58885)(3177) = 186683145 \)
- \( n(\sum x^2) = 8(415845139) = 3326761112 \)
- \( (58885)^2 = 3465869225 \)
So we have:
\[
m = \frac{90422080 - 186683145}{3326761112 - 3465869225}
\]
\[
m = \frac{-96261065}{-138108413} \approx 0.696
\]
2. **Calculate the intercept**:
\[
b = \frac{3177 - (0.696)(58885)}{8}
\]
Calculating this step-by-step:
- \( (0.696)(58885) \approx 41029.56 \)
- \( b = \frac{3177 - 41029.56}{8} \approx \frac{-37852.56}{8} \approx -4723.1325 \)
### Final Equation
Thus, rounding our slope and intercept to the nearest thousandth, we conclude:
\[
y \approx 0.696x - 4723.133
\]
So the equation for the least squares regression line is:
\[
\boxed{y = 0.696x - 4723.133}
\]
### Step 1: Prepare the Data
We are given the following pairs of data (students, organizations):
- (3098, 342)
- (3502, 204)
- (3666, 317)
- (5723, 377)
- (6583, 456)
- (7081, 478)
- (8356, 495)
- (9776, 498)
Let's denote our data:
\[
\text{Students } (x): [3098, 3502, 3666, 5723, 6583, 7081, 8356, 9776]
\]
\[
\text{Organizations } (y): [342, 204, 317, 377, 456, 478, 495, 498]
\]
### Step 2: Calculate Necessary Sums for Least Squares
We need the following sums for calculating the slope and intercept:
1. \( n = \) the number of data points
2. \( \sum x \) (sum of all x values)
3. \( \sum y \) (sum of all y values)
4. \( \sum xy \) (sum of the product of x and y)
5. \( \sum x^2 \) (sum of squares of x values)
Let's calculate these:
- \( n = 8 \)
- \( \sum x = 3098 + 3502 + 3666 + 5723 + 6583 + 7081 + 8356 + 9776 = 58885 \)
- \( \sum y = 342 + 204 + 317 + 377 + 456 + 478 + 495 + 498 = 3177 \)
- \( \sum xy = (3098 \cdot 342) + (3502 \cdot 204) + (3666 \cdot 317) + (5723 \cdot 377) + (6583 \cdot 456) + (7081 \cdot 478) + (8356 \cdot 495) + (9776 \cdot 498) = 11302760 \)
- \( \sum x^2 = (3098^2) + (3502^2) + (3666^2) + (5723^2) + (6583^2) + (7081^2) + (8356^2) + (9776^2) = 415845139 \)
### Step 3: Calculate the Slope (\( m \)) and Intercept (\( b \))
Using the formulas for the slope \( m \) and intercept \( b \):
\[
m = \frac{n(\sum xy) - (\sum x)(\sum y)}{n(\sum x^2) - (\sum x)^2}
\]
\[
b = \frac{\sum y - m(\sum x)}{n}
\]
Plugging in the values:
1. **Calculate the slope**:
\[
m = \frac{8(11302760) - (58885)(3177)}{8(415845139) - (58885)^2}
\]
Calculating step-by-step:
- \( 8(11302760) = 90422080 \)
- \( (58885)(3177) = 186683145 \)
- \( n(\sum x^2) = 8(415845139) = 3326761112 \)
- \( (58885)^2 = 3465869225 \)
So we have:
\[
m = \frac{90422080 - 186683145}{3326761112 - 3465869225}
\]
\[
m = \frac{-96261065}{-138108413} \approx 0.696
\]
2. **Calculate the intercept**:
\[
b = \frac{3177 - (0.696)(58885)}{8}
\]
Calculating this step-by-step:
- \( (0.696)(58885) \approx 41029.56 \)
- \( b = \frac{3177 - 41029.56}{8} \approx \frac{-37852.56}{8} \approx -4723.1325 \)
### Final Equation
Thus, rounding our slope and intercept to the nearest thousandth, we conclude:
\[
y \approx 0.696x - 4723.133
\]
So the equation for the least squares regression line is:
\[
\boxed{y = 0.696x - 4723.133}
\]