Question
Find the equation for the least squares regression line of the data described below.
Allie thinks that text messaging is causing her to talk less on the phone. For one month, she examined her text message and call logs with her closest friends.
For each friend, Allie checked the number of text messages she sent to that friend, x, and the number of minutes they spoke on the phone, y.
Text messages sent Minutes on the phone
137 241
174 250
311 277
461 169
475 123
488 176
Round your answers to the nearest thousandth.
y=
x+
Answers
Answered by
GPT-4o mini
To find the least squares regression line, we will need to perform some calculations to find the slope (m) and y-intercept (b) of the line. The formula for the least squares regression line is given by:
\[ y = mx + b \]
### Step 1: Gather the data points
We have the following data points:
- (137, 241)
- (174, 250)
- (311, 277)
- (461, 169)
- (475, 123)
- (488, 176)
### Step 2: Calculate the necessary sums
We need to calculate the following values:
1. \( n \) = 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 each x and y pair
5. \( \sum x^2 \) = sum of the squares of each x value
6. \( \sum y^2 \) = sum of the squares of each y value
Starting with the calculations:
- \( n = 6 \)
- **Sum of x values**:
\[
\sum x = 137 + 174 + 311 + 461 + 475 + 488 = 2046
\]
- **Sum of y values**:
\[
\sum y = 241 + 250 + 277 + 169 + 123 + 176 = 1236
\]
- **Sum of xy**:
\[
\sum xy = (137 \cdot 241) + (174 \cdot 250) + (311 \cdot 277) + (461 \cdot 169) + (475 \cdot 123) + (488 \cdot 176)
\]
Calculating each term:
\[
137 \cdot 241 = 32917
\]
\[
174 \cdot 250 = 43500
\]
\[
311 \cdot 277 = 86147
\]
\[
461 \cdot 169 = 77809
\]
\[
475 \cdot 123 = 58525
\]
\[
488 \cdot 176 = 85888
\]
Summing these gives:
\[
\sum xy = 32917 + 43500 + 86147 + 77809 + 58525 + 85888 = 288786
\]
- **Sum of x²**:
\[
\sum x^2 = 137^2 + 174^2 + 311^2 + 461^2 + 475^2 + 488^2
\]
Calculating each term:
\[
137^2 = 18769
\]
\[
174^2 = 30276
\]
\[
311^2 = 96721
\]
\[
461^2 = 212521
\]
\[
475^2 = 225625
\]
\[
488^2 = 238144
\]
Summing these gives:
\[
\sum x^2 = 18769 + 30276 + 96721 + 212521 + 225625 + 238144 = 800056
\]
### Step 3: Calculate the slope (m) and y-intercept (b)
The slope \( m \) is given by:
\[
m = \frac{n(\sum xy) - (\sum x)(\sum y)}{n(\sum x^2) - (\sum x)^2}
\]
Substituting the values:
\[
m = \frac{6(288786) - (2046)(1236)}{6(800056) - (2046)^2}
\]
Calculating the numerator:
\[
6(288786) = 1732736
\]
\[
(2046)(1236) = 2533056
\]
\[
\text{Numerator} = 1732736 - 2533056 = -800320
\]
Calculating the denominator:
\[
6(800056) = 4803360
\]
\[
(2046)^2 = 4189216
\]
\[
\text{Denominator} = 4803360 - 4189216 = 613144
\]
Now calculate \( m \):
\[
m = \frac{-800320}{613144} \approx -1.304
\]
Now we calculate the y-intercept \( b \):
\[
b = \frac{\sum y - m(\sum x)}{n}
\]
Calculating:
\[
b = \frac{1236 - (-1.304)(2046)}{6}
\]
\[
= \frac{1236 + 2668.384}{6} \approx \frac{3904.384}{6} \approx 650.731
\]
### Final Equation
The equation for the least squares regression line can be expressed as:
\[
y \approx -1.304x + 650.731
\]
Rounding to the nearest thousandth:
\[
y = -1.304x + 650.731
\]
\[ y = mx + b \]
### Step 1: Gather the data points
We have the following data points:
- (137, 241)
- (174, 250)
- (311, 277)
- (461, 169)
- (475, 123)
- (488, 176)
### Step 2: Calculate the necessary sums
We need to calculate the following values:
1. \( n \) = 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 each x and y pair
5. \( \sum x^2 \) = sum of the squares of each x value
6. \( \sum y^2 \) = sum of the squares of each y value
Starting with the calculations:
- \( n = 6 \)
- **Sum of x values**:
\[
\sum x = 137 + 174 + 311 + 461 + 475 + 488 = 2046
\]
- **Sum of y values**:
\[
\sum y = 241 + 250 + 277 + 169 + 123 + 176 = 1236
\]
- **Sum of xy**:
\[
\sum xy = (137 \cdot 241) + (174 \cdot 250) + (311 \cdot 277) + (461 \cdot 169) + (475 \cdot 123) + (488 \cdot 176)
\]
Calculating each term:
\[
137 \cdot 241 = 32917
\]
\[
174 \cdot 250 = 43500
\]
\[
311 \cdot 277 = 86147
\]
\[
461 \cdot 169 = 77809
\]
\[
475 \cdot 123 = 58525
\]
\[
488 \cdot 176 = 85888
\]
Summing these gives:
\[
\sum xy = 32917 + 43500 + 86147 + 77809 + 58525 + 85888 = 288786
\]
- **Sum of x²**:
\[
\sum x^2 = 137^2 + 174^2 + 311^2 + 461^2 + 475^2 + 488^2
\]
Calculating each term:
\[
137^2 = 18769
\]
\[
174^2 = 30276
\]
\[
311^2 = 96721
\]
\[
461^2 = 212521
\]
\[
475^2 = 225625
\]
\[
488^2 = 238144
\]
Summing these gives:
\[
\sum x^2 = 18769 + 30276 + 96721 + 212521 + 225625 + 238144 = 800056
\]
### Step 3: Calculate the slope (m) and y-intercept (b)
The slope \( m \) is given by:
\[
m = \frac{n(\sum xy) - (\sum x)(\sum y)}{n(\sum x^2) - (\sum x)^2}
\]
Substituting the values:
\[
m = \frac{6(288786) - (2046)(1236)}{6(800056) - (2046)^2}
\]
Calculating the numerator:
\[
6(288786) = 1732736
\]
\[
(2046)(1236) = 2533056
\]
\[
\text{Numerator} = 1732736 - 2533056 = -800320
\]
Calculating the denominator:
\[
6(800056) = 4803360
\]
\[
(2046)^2 = 4189216
\]
\[
\text{Denominator} = 4803360 - 4189216 = 613144
\]
Now calculate \( m \):
\[
m = \frac{-800320}{613144} \approx -1.304
\]
Now we calculate the y-intercept \( b \):
\[
b = \frac{\sum y - m(\sum x)}{n}
\]
Calculating:
\[
b = \frac{1236 - (-1.304)(2046)}{6}
\]
\[
= \frac{1236 + 2668.384}{6} \approx \frac{3904.384}{6} \approx 650.731
\]
### Final Equation
The equation for the least squares regression line can be expressed as:
\[
y \approx -1.304x + 650.731
\]
Rounding to the nearest thousandth:
\[
y = -1.304x + 650.731
\]
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