Asked by julian celeste
Use the table to answer the question.
x y
−5
3
−4
4
−3
7
−2
7.5
−1
2.5
Use a spreadsheet calculator to determine the least squares regression line for the points in the table. Round to the nearest hundredth if necessary.
(1 point)
Responses
y=5.55x+0.25
y equals 5.55 x plus 0.25
y=0.25x+5.55
y equals 0.25 x plus 5.55
y=0.25x−5.55
y equals 0.25 x minus 5.55
y=5.55x−0.25
x y
−5
3
−4
4
−3
7
−2
7.5
−1
2.5
Use a spreadsheet calculator to determine the least squares regression line for the points in the table. Round to the nearest hundredth if necessary.
(1 point)
Responses
y=5.55x+0.25
y equals 5.55 x plus 0.25
y=0.25x+5.55
y equals 0.25 x plus 5.55
y=0.25x−5.55
y equals 0.25 x minus 5.55
y=5.55x−0.25
Answers
There are no human answers yet.
Answered by
GPT-4o mini
To find the least squares regression line for the given points, we can perform the necessary calculations using the data provided.
First, we calculate the averages of \(x\) and \(y\):
1. Calculate the averages:
- \( \text{mean}(x) = \frac{-5 + (-4) + (-3) + (-2) + (-1)}{5} = \frac{-15}{5} = -3 \)
- \( \text{mean}(y) = \frac{3 + 4 + 7 + 7.5 + 2.5}{5} = \frac{24}{5} = 4.8 \)
2. Next, we compute the slope \(m\) using the formula:
\[
m = \frac{\sum((x_i - \bar{x})(y_i - \bar{y}))}{\sum((x_i - \bar{x})^2)}
\]
3. Calculate the necessary sums:
- \( \sum((x_i - \bar{x})(y_i - \bar{y})) \)
- \( \sum((x_i - \bar{x})^2) \)
4. For our dataset:
- \(x_i\) values: -5, -4, -3, -2, -1
- \(y_i\) values: 3, 4, 7, 7.5, 2.5
We calculate \(x_i - \bar{x}\) and \(y_i - \bar{y}\):
- For \(x = -5\), \(y = 3\): \((-5 - (-3))(3 - 4.8) = (-2)(-1.8) = 3.6\)
- For \(x = -4\), \(y = 4\): \((-4 - (-3))(4 - 4.8) = (-1)(-0.8) = 0.8\)
- For \(x = -3\), \(y = 7\): \((-3 - (-3))(7 - 4.8) = (0)(2.2) = 0\)
- For \(x = -2\), \(y = 7.5\): \((-2 - (-3))(7.5 - 4.8) = (1)(2.7) = 2.7\)
- For \(x = -1\), \(y = 2.5\): \((-1 - (-3))(2.5 - 4.8) = (2)(-2.3) = -4.6\)
So,
\[
\sum((x_i - \bar{x})(y_i - \bar{y})) = 3.6 + 0.8 + 0 + 2.7 - 4.6 = 2.5
\]
Now calculate \(\sum((x_i - \bar{x})^2)\):
- \((-5 + 3)^2 = 4\)
- \((-4 + 3)^2 = 1\)
- \((-3 + 3)^2 = 0\)
- \((-2 + 3)^2 = 1\)
- \((-1 + 3)^2 = 4\)
Adding these together:
\[
\sum((x_i - \bar{x})^2) = 4 + 1 + 0 + 1 + 4 = 10
\]
Now substituting into the slope formula:
\[
m = \frac{2.5}{10} = 0.25
\]
5. The y-intercept \(b\) is calculated using:
\[
b = \bar{y} - m \cdot \bar{x} = 4.8 - 0.25 \cdot (-3) = 4.8 + 0.75 = 5.55
\]
Now we have the equation of the line:
\[
y = mx + b = 0.25x + 5.55
\]
Thus the least squares regression line is:
\[
y = 0.25x + 5.55
\]
Among the given options:
- \(y = 0.25x + 5.55\)
is the correct answer.
First, we calculate the averages of \(x\) and \(y\):
1. Calculate the averages:
- \( \text{mean}(x) = \frac{-5 + (-4) + (-3) + (-2) + (-1)}{5} = \frac{-15}{5} = -3 \)
- \( \text{mean}(y) = \frac{3 + 4 + 7 + 7.5 + 2.5}{5} = \frac{24}{5} = 4.8 \)
2. Next, we compute the slope \(m\) using the formula:
\[
m = \frac{\sum((x_i - \bar{x})(y_i - \bar{y}))}{\sum((x_i - \bar{x})^2)}
\]
3. Calculate the necessary sums:
- \( \sum((x_i - \bar{x})(y_i - \bar{y})) \)
- \( \sum((x_i - \bar{x})^2) \)
4. For our dataset:
- \(x_i\) values: -5, -4, -3, -2, -1
- \(y_i\) values: 3, 4, 7, 7.5, 2.5
We calculate \(x_i - \bar{x}\) and \(y_i - \bar{y}\):
- For \(x = -5\), \(y = 3\): \((-5 - (-3))(3 - 4.8) = (-2)(-1.8) = 3.6\)
- For \(x = -4\), \(y = 4\): \((-4 - (-3))(4 - 4.8) = (-1)(-0.8) = 0.8\)
- For \(x = -3\), \(y = 7\): \((-3 - (-3))(7 - 4.8) = (0)(2.2) = 0\)
- For \(x = -2\), \(y = 7.5\): \((-2 - (-3))(7.5 - 4.8) = (1)(2.7) = 2.7\)
- For \(x = -1\), \(y = 2.5\): \((-1 - (-3))(2.5 - 4.8) = (2)(-2.3) = -4.6\)
So,
\[
\sum((x_i - \bar{x})(y_i - \bar{y})) = 3.6 + 0.8 + 0 + 2.7 - 4.6 = 2.5
\]
Now calculate \(\sum((x_i - \bar{x})^2)\):
- \((-5 + 3)^2 = 4\)
- \((-4 + 3)^2 = 1\)
- \((-3 + 3)^2 = 0\)
- \((-2 + 3)^2 = 1\)
- \((-1 + 3)^2 = 4\)
Adding these together:
\[
\sum((x_i - \bar{x})^2) = 4 + 1 + 0 + 1 + 4 = 10
\]
Now substituting into the slope formula:
\[
m = \frac{2.5}{10} = 0.25
\]
5. The y-intercept \(b\) is calculated using:
\[
b = \bar{y} - m \cdot \bar{x} = 4.8 - 0.25 \cdot (-3) = 4.8 + 0.75 = 5.55
\]
Now we have the equation of the line:
\[
y = mx + b = 0.25x + 5.55
\]
Thus the least squares regression line is:
\[
y = 0.25x + 5.55
\]
Among the given options:
- \(y = 0.25x + 5.55\)
is the correct answer.
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