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
1.4
1.7
2.5
3.3
9
4
8
5
4
Item S
<
(4 points)
Round numbers in the regression line to the nearest
thousandths.
x
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
1.4
1.7
2.5
3.3
9
4
8
5
4
Item S
<
(4 points)
Round numbers in the regression line to the nearest
thousandths.
x
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 determine how many hours of sleep Pierce would get if he was on his phone for 5 hours before bed, we first need to calculate the regression line based on the data you've provided.
Here’s the data in a clearer format:
| Time on Phone (x) | Hours of Sleep (y) |
|---------------------|---------------------|
| 0.4 | 9 |
| 1.4 | 8 |
| 1.7 | 5 |
| 2.5 | 4 |
| 3.3 | 4 |
Now let's calculate the regression line. The regression line can be expressed in the form:
\[ y = mx + b \]
where:
- \( m \) is the slope,
- \( b \) is the y-intercept,
- \( x \) is the independent variable (time on phone),
- \( y \) is the dependent variable (hours of sleep).
### Step 1: Calculate Means
First, we calculate the means of \( x \) and \( y \).
\[
\text{Mean of } x = \frac{0.4 + 1.4 + 1.7 + 2.5 + 3.3}{5} = \frac{9.0}{5} = 1.8
\]
\[
\text{Mean of } y = \frac{9 + 8 + 5 + 4 + 4}{5} = \frac{30}{5} = 6
\]
### Step 2: Calculate Slope (m)
The slope \( m \) is given by:
\[
m = \frac{n(\sum xy) - (\sum x)(\sum y)}{n(\sum x^2) - (\sum x)^2}
\]
We need to compute several values:
- \( n = 5 \) (number of data points)
- \( \sum x = 0.4 + 1.4 + 1.7 + 2.5 + 3.3 = 9.0 \)
- \( \sum y = 9 + 8 + 5 + 4 + 4 = 30 \)
- \( \sum xy = (0.4 * 9) + (1.4 * 8) + (1.7 * 5) + (2.5 * 4) + (3.3 * 4) = 3.6 + 11.2 + 8.5 + 10 + 13.2 = 46.5 \)
- \( \sum x^2 = (0.4^2) + (1.4^2) + (1.7^2) + (2.5^2) + (3.3^2) = 0.16 + 1.96 + 2.89 + 6.25 + 10.89 = 22.15 \)
Now substituting these values into the equation for \( m \):
\[
m = \frac{5(46.5) - (9)(30)}{5(22.15) - (9)^2}
\]
Calculating:
\[
m = \frac{232.5 - 270}{110.75 - 81} = \frac{-37.5}{29.75} \approx -1.26
\]
### Step 3: Calculate Intercept (b)
The y-intercept \( b \) is calculated as follows:
\[
b = \text{Mean of } y - m(\text{Mean of } x)
\]
So,
\[
b = 6 - (-1.26)(1.8) = 6 + 2.268 \approx 8.268
\]
### Step 4: Regression Equation
Putting the slope and intercept together:
\[
y \approx -1.26x + 8.268
\]
### Step 5: Predicting Hours of Sleep with 5 Hours on Phone
Now we can substitute \( x = 5 \) into the regression equation.
\[
y \approx -1.26(5) + 8.268 \approx -6.3 + 8.268 \approx 1.968
\]
Rounding to the nearest thousandths, the prediction of hours of sleep is:
**1.968 hours**
So if Pierce spends 5 hours on his phone before bed, he would be expected to get approximately **1.968 hours of sleep**.
Here’s the data in a clearer format:
| Time on Phone (x) | Hours of Sleep (y) |
|---------------------|---------------------|
| 0.4 | 9 |
| 1.4 | 8 |
| 1.7 | 5 |
| 2.5 | 4 |
| 3.3 | 4 |
Now let's calculate the regression line. The regression line can be expressed in the form:
\[ y = mx + b \]
where:
- \( m \) is the slope,
- \( b \) is the y-intercept,
- \( x \) is the independent variable (time on phone),
- \( y \) is the dependent variable (hours of sleep).
### Step 1: Calculate Means
First, we calculate the means of \( x \) and \( y \).
\[
\text{Mean of } x = \frac{0.4 + 1.4 + 1.7 + 2.5 + 3.3}{5} = \frac{9.0}{5} = 1.8
\]
\[
\text{Mean of } y = \frac{9 + 8 + 5 + 4 + 4}{5} = \frac{30}{5} = 6
\]
### Step 2: Calculate Slope (m)
The slope \( m \) is given by:
\[
m = \frac{n(\sum xy) - (\sum x)(\sum y)}{n(\sum x^2) - (\sum x)^2}
\]
We need to compute several values:
- \( n = 5 \) (number of data points)
- \( \sum x = 0.4 + 1.4 + 1.7 + 2.5 + 3.3 = 9.0 \)
- \( \sum y = 9 + 8 + 5 + 4 + 4 = 30 \)
- \( \sum xy = (0.4 * 9) + (1.4 * 8) + (1.7 * 5) + (2.5 * 4) + (3.3 * 4) = 3.6 + 11.2 + 8.5 + 10 + 13.2 = 46.5 \)
- \( \sum x^2 = (0.4^2) + (1.4^2) + (1.7^2) + (2.5^2) + (3.3^2) = 0.16 + 1.96 + 2.89 + 6.25 + 10.89 = 22.15 \)
Now substituting these values into the equation for \( m \):
\[
m = \frac{5(46.5) - (9)(30)}{5(22.15) - (9)^2}
\]
Calculating:
\[
m = \frac{232.5 - 270}{110.75 - 81} = \frac{-37.5}{29.75} \approx -1.26
\]
### Step 3: Calculate Intercept (b)
The y-intercept \( b \) is calculated as follows:
\[
b = \text{Mean of } y - m(\text{Mean of } x)
\]
So,
\[
b = 6 - (-1.26)(1.8) = 6 + 2.268 \approx 8.268
\]
### Step 4: Regression Equation
Putting the slope and intercept together:
\[
y \approx -1.26x + 8.268
\]
### Step 5: Predicting Hours of Sleep with 5 Hours on Phone
Now we can substitute \( x = 5 \) into the regression equation.
\[
y \approx -1.26(5) + 8.268 \approx -6.3 + 8.268 \approx 1.968
\]
Rounding to the nearest thousandths, the prediction of hours of sleep is:
**1.968 hours**
So if Pierce spends 5 hours on his phone before bed, he would be expected to get approximately **1.968 hours of sleep**.