Question
Find the equation for the least squares regression line of the data described below. use desmos
The Harrell toy company is testing the usability of its new spinning top among different age groups. Caleb, a designer at the company, gave the tops to a group of children of different ages and had each child spin a top once.
Caleb recorded the age of each child, x, and how long he or she got the top to spin (in seconds), y.
Age Time (in seconds)
3 5
3 15
4 12
5 86
8 62
8 77
Round your answers to the nearest thousandth.
y=
x+
Answers
GPT-4o mini
To find the least squares regression line for the given data, we first need to outline the provided data points clearly:
- (3, 5)
- (3, 15)
- (4, 12)
- (5, 86)
- (8, 62)
- (8, 77)
**Step 1: Calculating the necessary sums for the regression line.**
We will find the required sums for the calculations:
1. **Calculate the means of x and y:**
\[
\bar{x} = \frac{3 + 3 + 4 + 5 + 8 + 8}{6} = \frac{31}{6} \approx 5.167
\]
\[
\bar{y} = \frac{5 + 15 + 12 + 86 + 62 + 77}{6} = \frac{257}{6} \approx 42.833
\]
2. **Calculate the sums:**
For \( S_{xx} \) and \( S_{xy} \):
\[
S_{xx} = \sum (x_i - \bar{x})^2 = (3 - 5.167)^2 + (3 - 5.167)^2 + (4 - 5.167)^2 + (5 - 5.167)^2 + (8 - 5.167)^2 + (8 - 5.167)^2
\]
Calculating these values:
\[
S_{xx} = (-2.167)^2 + (-2.167)^2 + (-1.167)^2 + (-0.167)^2 + (2.833)^2 + (2.833)^2
\]
\[
S_{xx} = 4.696 + 4.696 + 1.362 + 0.028 + 8.027 + 8.027 \approx 26.836
\]
For \( S_{xy} \):
\[
S_{xy} = \sum (x_i - \bar{x})(y_i - \bar{y}) = (3 - 5.167)(5 - 42.833) + (3 - 5.167)(15 - 42.833) + (4 - 5.167)(12 - 42.833) + (5 - 5.167)(86 - 42.833) + (8 - 5.167)(62 - 42.833) + (8 - 5.167)(77 - 42.833)
\]
Computing this gives:
1. \( (3 - 5.167)(5 - 42.833) \approx 2.167 \times 37.833 \approx 82.036 \)
2. \( (3 - 5.167)(15 - 42.833) \approx 2.167 \times 27.833 \approx 60.49 \)
3. \( (4 - 5.167)(12 - 42.833) = (-1.167)(-30.833) \approx 36.03\)
4. \( (5 - 5.167)(86 - 42.833) = (-0.167)(43.167) \approx -7.19\)
5. \( (8 - 5.167)(62 - 42.833) \approx 2.833 \times 19.167 \approx 54.267\)
6. \( (8 - 5.167)(77 - 42.833) \approx 2.833 \times 34.167 \approx 96.79\)
Adding those values yields:
\[
S_{xy} \approx 82.036 + 60.49 + 36.03 - 7.19 + 54.267 + 96.79 \approx 322.42
\]
**Step 2: Calculating the slope (m) and intercept (b).**
The slope \( m \) is:
\[
m = \frac{S_{xy}}{S_{xx}} \approx \frac{322.42}{26.836} \approx 12.006
\]
The y-intercept \( b \) is given by:
\[
b = \bar{y} - m\bar{x} \approx 42.833 - 12.006 \times 5.167 \approx 42.833 - 61.95 \approx -19.117
\]
**Step 3: Formulating the model.**
Thus, the least squares regression line equation is:
\[
y \approx 12.006x - 19.117
\]
**Answer:**
After rounding, the equation becomes:
\[
y = 12.006x - 19.117
\]
So the final answer is:
\[
y = 12.006x - 19.117
\]
- (3, 5)
- (3, 15)
- (4, 12)
- (5, 86)
- (8, 62)
- (8, 77)
**Step 1: Calculating the necessary sums for the regression line.**
We will find the required sums for the calculations:
1. **Calculate the means of x and y:**
\[
\bar{x} = \frac{3 + 3 + 4 + 5 + 8 + 8}{6} = \frac{31}{6} \approx 5.167
\]
\[
\bar{y} = \frac{5 + 15 + 12 + 86 + 62 + 77}{6} = \frac{257}{6} \approx 42.833
\]
2. **Calculate the sums:**
For \( S_{xx} \) and \( S_{xy} \):
\[
S_{xx} = \sum (x_i - \bar{x})^2 = (3 - 5.167)^2 + (3 - 5.167)^2 + (4 - 5.167)^2 + (5 - 5.167)^2 + (8 - 5.167)^2 + (8 - 5.167)^2
\]
Calculating these values:
\[
S_{xx} = (-2.167)^2 + (-2.167)^2 + (-1.167)^2 + (-0.167)^2 + (2.833)^2 + (2.833)^2
\]
\[
S_{xx} = 4.696 + 4.696 + 1.362 + 0.028 + 8.027 + 8.027 \approx 26.836
\]
For \( S_{xy} \):
\[
S_{xy} = \sum (x_i - \bar{x})(y_i - \bar{y}) = (3 - 5.167)(5 - 42.833) + (3 - 5.167)(15 - 42.833) + (4 - 5.167)(12 - 42.833) + (5 - 5.167)(86 - 42.833) + (8 - 5.167)(62 - 42.833) + (8 - 5.167)(77 - 42.833)
\]
Computing this gives:
1. \( (3 - 5.167)(5 - 42.833) \approx 2.167 \times 37.833 \approx 82.036 \)
2. \( (3 - 5.167)(15 - 42.833) \approx 2.167 \times 27.833 \approx 60.49 \)
3. \( (4 - 5.167)(12 - 42.833) = (-1.167)(-30.833) \approx 36.03\)
4. \( (5 - 5.167)(86 - 42.833) = (-0.167)(43.167) \approx -7.19\)
5. \( (8 - 5.167)(62 - 42.833) \approx 2.833 \times 19.167 \approx 54.267\)
6. \( (8 - 5.167)(77 - 42.833) \approx 2.833 \times 34.167 \approx 96.79\)
Adding those values yields:
\[
S_{xy} \approx 82.036 + 60.49 + 36.03 - 7.19 + 54.267 + 96.79 \approx 322.42
\]
**Step 2: Calculating the slope (m) and intercept (b).**
The slope \( m \) is:
\[
m = \frac{S_{xy}}{S_{xx}} \approx \frac{322.42}{26.836} \approx 12.006
\]
The y-intercept \( b \) is given by:
\[
b = \bar{y} - m\bar{x} \approx 42.833 - 12.006 \times 5.167 \approx 42.833 - 61.95 \approx -19.117
\]
**Step 3: Formulating the model.**
Thus, the least squares regression line equation is:
\[
y \approx 12.006x - 19.117
\]
**Answer:**
After rounding, the equation becomes:
\[
y = 12.006x - 19.117
\]
So the final answer is:
\[
y = 12.006x - 19.117
\]