Question
Pierce isn't getting enough sleep and suspects it is due to spending too much time on his phone before bed. Over the next several days, he decided to track how much time he spends on his phone, x, and how many hours of sleep he gets, y.
Hours on the phone before bed | Hours of sleep
0.4 | 9
1.4 | 4
1.7 | 8
2.5 | 5
3.3 | 4
Round numbers in the regression line to the nearest thousandths.
r = BLANK
y = BLANKx + BLANK
Using this regression line, how many hours of sleep would Pierce get if he was on his phone for 5 hours before bed?
Hours on the phone before bed | Hours of sleep
0.4 | 9
1.4 | 4
1.7 | 8
2.5 | 5
3.3 | 4
Round numbers in the regression line to the nearest thousandths.
r = BLANK
y = BLANKx + BLANK
Using this regression line, how many hours of sleep would Pierce get if he was on his phone for 5 hours before bed?
Answers
GPT-4o mini
To analyze Pierce's data, we can calculate the linear regression line, which has the form:
\[ y = mx + b \]
where \( y \) is the number of hours of sleep, \( x \) is the hours on the phone, \( m \) is the slope, and \( b \) is the y-intercept.
Given the points:
- (0.4, 9)
- (1.4, 4)
- (1.7, 8)
- (2.5, 5)
- (3.3, 4)
### Step 1: Calculate the slope (m) and y-intercept (b)
1. **Calculate the means** of \( x \) and \( y \):
\[
\bar{x} = \frac{0.4 + 1.4 + 1.7 + 2.5 + 3.3}{5} = \frac{9.0}{5} = 1.8
\]
\[
\bar{y} = \frac{9 + 4 + 8 + 5 + 4}{5} = \frac{30}{5} = 6
\]
2. **Calculate the slope (m)**:
\[
m = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sum (x_i - \bar{x})^2}
\]
- Calculate \( \sum (x_i - \bar{x})(y_i - \bar{y}) \):
\[
(0.4 - 1.8)(9 - 6) + (1.4 - 1.8)(4 - 6) + (1.7 - 1.8)(8 - 6) + (2.5 - 1.8)(5 - 6) + (3.3 - 1.8)(4 - 6)
\]
\[
= (-1.4)(3) + (-0.4)(-2) + (-0.1)(2) + (0.7)(-1) + (1.5)(-2)
\]
\[
= -4.2 + 0.8 - 0.2 - 0.7 - 3.0 = -7.3
\]
- Calculate \( \sum (x_i - \bar{x})^2 \):
\[
(0.4 - 1.8)^2 + (1.4 - 1.8)^2 + (1.7 - 1.8)^2 + (2.5 - 1.8)^2 + (3.3 - 1.8)^2
\]
\[
= (-1.4)^2 + (-0.4)^2 + (-0.1)^2 + (0.7)^2 + (1.5)^2
\]
\[
= 1.96 + 0.16 + 0.01 + 0.49 + 2.25 = 4.87
\]
- Calculate the slope:
\[
m = \frac{-7.3}{4.87} \approx -1.498
\]
3. **Calculate the y-intercept (b)**:
\[
b = \bar{y} - m\bar{x}
\]
\[
b = 6 - (-1.498)(1.8) \approx 6 + 2.6964 \approx 8.696
\]
### Step 2: Writing the regression line
Now we round the numbers to the nearest thousandths:
- Slope \( m \approx -1.498 \)
- y-intercept \( b \approx 8.696 \)
The regression line is given by:
\[
y = -1.498x + 8.696
\]
### Step 3: Predicting sleep for \( x = 5 \) hours on the phone
Using the regression line:
\[
y = -1.498(5) + 8.696
\]
\[
y = -7.49 + 8.696 \approx 1.206
\]
### Final Answers
- \( r \approx -1.498 \)
- \( y = -1.498x + 8.696 \)
If Pierce was on his phone for 5 hours before bed, he would get approximately **1.206 hours of sleep**.
\[ y = mx + b \]
where \( y \) is the number of hours of sleep, \( x \) is the hours on the phone, \( m \) is the slope, and \( b \) is the y-intercept.
Given the points:
- (0.4, 9)
- (1.4, 4)
- (1.7, 8)
- (2.5, 5)
- (3.3, 4)
### Step 1: Calculate the slope (m) and y-intercept (b)
1. **Calculate the means** of \( x \) and \( y \):
\[
\bar{x} = \frac{0.4 + 1.4 + 1.7 + 2.5 + 3.3}{5} = \frac{9.0}{5} = 1.8
\]
\[
\bar{y} = \frac{9 + 4 + 8 + 5 + 4}{5} = \frac{30}{5} = 6
\]
2. **Calculate the slope (m)**:
\[
m = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sum (x_i - \bar{x})^2}
\]
- Calculate \( \sum (x_i - \bar{x})(y_i - \bar{y}) \):
\[
(0.4 - 1.8)(9 - 6) + (1.4 - 1.8)(4 - 6) + (1.7 - 1.8)(8 - 6) + (2.5 - 1.8)(5 - 6) + (3.3 - 1.8)(4 - 6)
\]
\[
= (-1.4)(3) + (-0.4)(-2) + (-0.1)(2) + (0.7)(-1) + (1.5)(-2)
\]
\[
= -4.2 + 0.8 - 0.2 - 0.7 - 3.0 = -7.3
\]
- Calculate \( \sum (x_i - \bar{x})^2 \):
\[
(0.4 - 1.8)^2 + (1.4 - 1.8)^2 + (1.7 - 1.8)^2 + (2.5 - 1.8)^2 + (3.3 - 1.8)^2
\]
\[
= (-1.4)^2 + (-0.4)^2 + (-0.1)^2 + (0.7)^2 + (1.5)^2
\]
\[
= 1.96 + 0.16 + 0.01 + 0.49 + 2.25 = 4.87
\]
- Calculate the slope:
\[
m = \frac{-7.3}{4.87} \approx -1.498
\]
3. **Calculate the y-intercept (b)**:
\[
b = \bar{y} - m\bar{x}
\]
\[
b = 6 - (-1.498)(1.8) \approx 6 + 2.6964 \approx 8.696
\]
### Step 2: Writing the regression line
Now we round the numbers to the nearest thousandths:
- Slope \( m \approx -1.498 \)
- y-intercept \( b \approx 8.696 \)
The regression line is given by:
\[
y = -1.498x + 8.696
\]
### Step 3: Predicting sleep for \( x = 5 \) hours on the phone
Using the regression line:
\[
y = -1.498(5) + 8.696
\]
\[
y = -7.49 + 8.696 \approx 1.206
\]
### Final Answers
- \( r \approx -1.498 \)
- \( y = -1.498x + 8.696 \)
If Pierce was on his phone for 5 hours before bed, he would get approximately **1.206 hours of sleep**.