Question
Use the table to answer the question.
x y
0 8
4 9
8 10
12 11
16 12
Raul uses a spreadsheet calculator to find the line of least squares regression for the data in the table. What is the equation for this line? Round to the nearest hundredth, if necessary.
(1 point)
Responses
y=0.25x+8
y is equal to 0 point 2 5 x plus 8 - no response given
y=8x+0.25
y is equal to 8 x plus 0 point 2 5 - no response given
y=−0.25x+8
y is equal to negative 0 point 2 5 x plus 8 - incorrect
y=0.25x−8
x y
0 8
4 9
8 10
12 11
16 12
Raul uses a spreadsheet calculator to find the line of least squares regression for the data in the table. What is the equation for this line? Round to the nearest hundredth, if necessary.
(1 point)
Responses
y=0.25x+8
y is equal to 0 point 2 5 x plus 8 - no response given
y=8x+0.25
y is equal to 8 x plus 0 point 2 5 - no response given
y=−0.25x+8
y is equal to negative 0 point 2 5 x plus 8 - incorrect
y=0.25x−8
Answers
There are no human answers yet.
Answered by
GPT-4o mini
To find the line of least squares regression for the given data, we can use the method of linear regression to derive the equation of the form \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.
Using the data provided:
\[
\begin{array}{c|c}
x & y \\
\hline
0 & 8 \\
4 & 9 \\
8 & 10 \\
12 & 11 \\
16 & 12 \\
\end{array}
\]
1. **Calculate the means of x and y:**
\( \bar{x} = \frac{0 + 4 + 8 + 12 + 16}{5} = \frac{40}{5} = 8 \)
\( \bar{y} = \frac{8 + 9 + 10 + 11 + 12}{5} = \frac{50}{5} = 10 \)
2. **Calculate \( m \) (the slope):**
\[
m = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sum (x_i - \bar{x})^2}
\]
- **Calculate the necessary values:**
\[
\begin{array}{c|c|c|c|c}
x & y & x - \bar{x} & y - \bar{y} & (x - \bar{x})(y - \bar{y}) \\
\hline
0 & 8 & 0 - 8 = -8 & 8 - 10 = -2 & 16 \\
4 & 9 & 4 - 8 = -4 & 9 - 10 = -1 & 4 \\
8 & 10 & 8 - 8 = 0 & 10 - 10 = 0 & 0 \\
12 & 11 & 12 - 8 = 4 & 11 - 10 = 1 & 4 \\
16 & 12 & 16 - 8 = 8 & 12 - 10 = 2 & 16 \\
\end{array}
\]
- **Sum of products:**
\[
\sum (x_i - \bar{x})(y_i - \bar{y}) = 16 + 4 + 0 + 4 + 16 = 40
\]
- **Sum of squares:**
\[
\sum (x_i - \bar{x})^2 = (-8)^2 + (-4)^2 + 0^2 + 4^2 + 8^2 = 64 + 16 + 0 + 16 + 64 = 160
\]
- **Calculate slope \( m \):**
\[
m = \frac{40}{160} = 0.25
\]
3. **Calculate \( b \) (the y-intercept):**
\[
b = \bar{y} - m\bar{x} = 10 - 0.25 \cdot 8 = 10 - 2 = 8
\]
4. **Final equation:**
Thus, the equation of the regression line is:
\[
y = 0.25x + 8
\]
Therefore, the correct response is:
**y = 0.25x + 8**.
Using the data provided:
\[
\begin{array}{c|c}
x & y \\
\hline
0 & 8 \\
4 & 9 \\
8 & 10 \\
12 & 11 \\
16 & 12 \\
\end{array}
\]
1. **Calculate the means of x and y:**
\( \bar{x} = \frac{0 + 4 + 8 + 12 + 16}{5} = \frac{40}{5} = 8 \)
\( \bar{y} = \frac{8 + 9 + 10 + 11 + 12}{5} = \frac{50}{5} = 10 \)
2. **Calculate \( m \) (the slope):**
\[
m = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sum (x_i - \bar{x})^2}
\]
- **Calculate the necessary values:**
\[
\begin{array}{c|c|c|c|c}
x & y & x - \bar{x} & y - \bar{y} & (x - \bar{x})(y - \bar{y}) \\
\hline
0 & 8 & 0 - 8 = -8 & 8 - 10 = -2 & 16 \\
4 & 9 & 4 - 8 = -4 & 9 - 10 = -1 & 4 \\
8 & 10 & 8 - 8 = 0 & 10 - 10 = 0 & 0 \\
12 & 11 & 12 - 8 = 4 & 11 - 10 = 1 & 4 \\
16 & 12 & 16 - 8 = 8 & 12 - 10 = 2 & 16 \\
\end{array}
\]
- **Sum of products:**
\[
\sum (x_i - \bar{x})(y_i - \bar{y}) = 16 + 4 + 0 + 4 + 16 = 40
\]
- **Sum of squares:**
\[
\sum (x_i - \bar{x})^2 = (-8)^2 + (-4)^2 + 0^2 + 4^2 + 8^2 = 64 + 16 + 0 + 16 + 64 = 160
\]
- **Calculate slope \( m \):**
\[
m = \frac{40}{160} = 0.25
\]
3. **Calculate \( b \) (the y-intercept):**
\[
b = \bar{y} - m\bar{x} = 10 - 0.25 \cdot 8 = 10 - 2 = 8
\]
4. **Final equation:**
Thus, the equation of the regression line is:
\[
y = 0.25x + 8
\]
Therefore, the correct response is:
**y = 0.25x + 8**.
Submit Your Answer
We prioritize human answers over AI answers.
If you are human, and you can answer this question, please submit your answer.