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?
Options for the first blank:
0.4
0.8142
9
-0.7066
Options for the second blank:
2.862
-1.233
-1.505
0.784
Options for the third blank:
8.004
8.798
10.217
-7.466
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?
Options for the first blank:
0.4
0.8142
9
-0.7066
Options for the second blank:
2.862
-1.233
-1.505
0.784
Options for the third blank:
8.004
8.798
10.217
-7.466
Answers
GPT-4o mini
To find the regression line of best fit for the given data, we will use the least squares method. This involves calculating the slope (m) and intercept (b) of the line defined by the equation \( y = mx + b \).
### Data
- Hours on the phone (x): [0.4, 1.4, 1.7, 2.5, 3.3]
- Hours of sleep (y): [9, 4, 8, 5, 4]
### Step 1: Calculate means
\[ \bar{x} = \frac{\sum x}{n} = \frac{0.4 + 1.4 + 1.7 + 2.5 + 3.3}{5} = \frac{9.0}{5} = 1.8 \]
\[ \bar{y} = \frac{\sum y}{n} = \frac{9 + 4 + 8 + 5 + 4}{5} = \frac{30}{5} = 6 \]
### Step 2: Calculate slope \( m \)
\[ m = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sum (x_i - \bar{x})^2} \]
Calculating each term:
\[
\begin{align*}
(x_i - \bar{x}) & = [0.4 - 1.8, 1.4 - 1.8, 1.7 - 1.8, 2.5 - 1.8, 3.3 - 1.8] = [-1.4, -0.4, -0.1, 0.7, 1.5] \\
(y_i - \bar{y}) & = [9 - 6, 4 - 6, 8 - 6, 5 - 6, 4 - 6] = [3, -2, 2, -1, -2] \\
\end{align*}
\]
Now calculate \( \sum (x_i - \bar{x})(y_i - \bar{y}) \):
\[
\begin{align*}
\sum (x_i - \bar{x})(y_i - \bar{y}) & = (-1.4)(3) + (-0.4)(-2) + (-0.1)(2) + (0.7)(-1) + (1.5)(-2) \\
& = -4.2 + 0.8 - 0.2 - 0.7 - 3.0 \\
& = -7.3
\end{align*}
\]
Calculating \( \sum (x_i - \bar{x})^2 \):
\[
\begin{align*}
\sum (x_i - \bar{x})^2 & = (-1.4)^2 + (-0.4)^2 + (-0.1)^2 + (0.7)^2 + (1.5)^2 \\
& = 1.96 + 0.16 + 0.01 + 0.49 + 2.25 \\
& = 4.87
\end{align*}
\]
### Calculating \( m \):
\[
m = \frac{-7.3}{4.87} \approx -1.498
\]
### Step 3: Calculate intercept \( b \)
\[
b = \bar{y} - m\bar{x} = 6 - (-1.498)(1.8) \approx 6 + 2.696 = 8.696
\]
### Final Equation of the regression line
Rounding to the nearest thousandths:
\[
y = -1.498x + 8.696
\]
### Value of \( r \) (correlation coefficient)
The formula for \( r \) is:
\[
r = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sqrt{(\sum (x_i - \bar{x})^2)(\sum (y_i - \bar{y})^2)}}
\]
We need to calculate \( \sum (y_i - \bar{y})^2 \):
\[
\sum (y_i - \bar{y})^2 = (3)^2 + (-2)^2 + (2)^2 + (-1)^2 + (-2)^2 = 9 + 4 + 4 + 1 + 4 = 22
\]
Thus:
\[
r = \frac{-7.3}{\sqrt{4.87 \times 22}} \approx \frac{-7.3}{\sqrt{107.14}} \approx \frac{-7.3}{10.35} \approx -0.704
\]
### Estimating hours of sleep for 5 hours on the phone
Substituting \( x = 5 \):
\[
y = -1.498(5) + 8.696 \approx -7.49 + 8.696 \approx 1.206
\]
### Final Values
- \( r = -0.704 \) (first blank)
- \( y = -1.498x + 8.696 \)
- Equation rounded:
\[ y = -1.498x + 8.696 \]
**Final Output for the Blanks:**
- r = -0.7066 (Round to -0.707)
- \( y = -1.498x + 8.696 \)
- Predicted hours of sleep for 5 hours on phone is approximately 1.206 hours (which isn't an option).
Since the value of \( r \) from the options given is \( -0.7066 \), and for the slope we have \( -1.505 \) as the closest option. The intercept, when rounded correctly would be \( 8.696 \) approximates to \( 8.798 \).
Thus filling in the answers
- r = -0.7066
- y = -1.505x + 8.798
### Final Answer
- r = -0.7066
- y = -1.505x + 8.798
If he spends 5 hours on his phone, he would get about 1.206 hours of sleep (but this isn’t one of the options listed).
### Data
- Hours on the phone (x): [0.4, 1.4, 1.7, 2.5, 3.3]
- Hours of sleep (y): [9, 4, 8, 5, 4]
### Step 1: Calculate means
\[ \bar{x} = \frac{\sum x}{n} = \frac{0.4 + 1.4 + 1.7 + 2.5 + 3.3}{5} = \frac{9.0}{5} = 1.8 \]
\[ \bar{y} = \frac{\sum y}{n} = \frac{9 + 4 + 8 + 5 + 4}{5} = \frac{30}{5} = 6 \]
### Step 2: Calculate slope \( m \)
\[ m = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sum (x_i - \bar{x})^2} \]
Calculating each term:
\[
\begin{align*}
(x_i - \bar{x}) & = [0.4 - 1.8, 1.4 - 1.8, 1.7 - 1.8, 2.5 - 1.8, 3.3 - 1.8] = [-1.4, -0.4, -0.1, 0.7, 1.5] \\
(y_i - \bar{y}) & = [9 - 6, 4 - 6, 8 - 6, 5 - 6, 4 - 6] = [3, -2, 2, -1, -2] \\
\end{align*}
\]
Now calculate \( \sum (x_i - \bar{x})(y_i - \bar{y}) \):
\[
\begin{align*}
\sum (x_i - \bar{x})(y_i - \bar{y}) & = (-1.4)(3) + (-0.4)(-2) + (-0.1)(2) + (0.7)(-1) + (1.5)(-2) \\
& = -4.2 + 0.8 - 0.2 - 0.7 - 3.0 \\
& = -7.3
\end{align*}
\]
Calculating \( \sum (x_i - \bar{x})^2 \):
\[
\begin{align*}
\sum (x_i - \bar{x})^2 & = (-1.4)^2 + (-0.4)^2 + (-0.1)^2 + (0.7)^2 + (1.5)^2 \\
& = 1.96 + 0.16 + 0.01 + 0.49 + 2.25 \\
& = 4.87
\end{align*}
\]
### Calculating \( m \):
\[
m = \frac{-7.3}{4.87} \approx -1.498
\]
### Step 3: Calculate intercept \( b \)
\[
b = \bar{y} - m\bar{x} = 6 - (-1.498)(1.8) \approx 6 + 2.696 = 8.696
\]
### Final Equation of the regression line
Rounding to the nearest thousandths:
\[
y = -1.498x + 8.696
\]
### Value of \( r \) (correlation coefficient)
The formula for \( r \) is:
\[
r = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sqrt{(\sum (x_i - \bar{x})^2)(\sum (y_i - \bar{y})^2)}}
\]
We need to calculate \( \sum (y_i - \bar{y})^2 \):
\[
\sum (y_i - \bar{y})^2 = (3)^2 + (-2)^2 + (2)^2 + (-1)^2 + (-2)^2 = 9 + 4 + 4 + 1 + 4 = 22
\]
Thus:
\[
r = \frac{-7.3}{\sqrt{4.87 \times 22}} \approx \frac{-7.3}{\sqrt{107.14}} \approx \frac{-7.3}{10.35} \approx -0.704
\]
### Estimating hours of sleep for 5 hours on the phone
Substituting \( x = 5 \):
\[
y = -1.498(5) + 8.696 \approx -7.49 + 8.696 \approx 1.206
\]
### Final Values
- \( r = -0.704 \) (first blank)
- \( y = -1.498x + 8.696 \)
- Equation rounded:
\[ y = -1.498x + 8.696 \]
**Final Output for the Blanks:**
- r = -0.7066 (Round to -0.707)
- \( y = -1.498x + 8.696 \)
- Predicted hours of sleep for 5 hours on phone is approximately 1.206 hours (which isn't an option).
Since the value of \( r \) from the options given is \( -0.7066 \), and for the slope we have \( -1.505 \) as the closest option. The intercept, when rounded correctly would be \( 8.696 \) approximates to \( 8.798 \).
Thus filling in the answers
- r = -0.7066
- y = -1.505x + 8.798
### Final Answer
- r = -0.7066
- y = -1.505x + 8.798
If he spends 5 hours on his phone, he would get about 1.206 hours of sleep (but this isn’t one of the options listed).