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Use the table to answer the question.

x y
−5
−
5
3
−4
−
4
4
−3
−
3
7
−2
−
2
7.5
−1
−
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
=
5.55
𝑥
−
0.25
y equals 5.55 x minus 0.25

y=0.25x−5.55
𝑦
=
0.25
𝑥
−
5.55

1 answer

To find the least squares regression line for the points in the table, we can use the formula for the linear regression line, which takes the form:

\[ y = mx + b \]

where \( m \) is the slope and \( b \) is the y-intercept.

Let's assume you have the data points:

  • (-5, 3)
  • (-4, 4)
  • (-3, 7)
  • (-2, 7.5)
  • (-1, 2.5)

Steps to Calculate the Least Squares Regression Line

  1. Calculate the means of \( x \) and \( y \): \[ \bar{x} = \frac{\sum{x}}{n}, \quad \bar{y} = \frac{\sum{y}}{n} \]

  2. Calculate the slope \( m \): \[ m = \frac{n(\sum{xy}) - (\sum{x})(\sum{y})}{n(\sum{x^2}) - (\sum{x})^2} \]

  3. Calculate the y-intercept \( b \): \[ b = \bar{y} - m\bar{x} \]

Example Calculation

Let’s calculate the necessary values from our dataset.

  1. Calculate the sums:

    • \( \sum{x} = -5 + -4 + -3 + -2 + -1 = -15 \)
    • \( \sum{y} = 3 + 4 + 7 + 7.5 + 2.5 = 24 \)
    • \( \sum{xy} = (-5)(3) + (-4)(4) + (-3)(7) + (-2)(7.5) + (-1)(2.5) = -15 - 16 - 21 - 15 + -2.5 = -69.5\)
    • \( \sum{x^2} = (-5)^2 + (-4)^2 + (-3)^2 + (-2)^2 + (-1)^2 = 25 + 16 + 9 + 4 + 1 = 55 \)
    • \( n = 5 \) (the number of points)

  2. Calculate the means: \[ \bar{x} = \frac{-15}{5} = -3, \quad \bar{y} = \frac{24}{5} = 4.8 \]

  3. Calculate slope \( m \): \[ m = \frac{5(-69.5) - (-15)(24)}{5(55) - (-15)^2} = \frac{-347.5 + 360}{275 - 225} = \frac{12.5}{50} = 0.25 \]

  4. Calculate y-intercept \( b \): \[ b = \bar{y} - m\bar{x} = 4.8 - 0.25(-3) = 4.8 + 0.75 = 5.55 \]

Final Equation

Thus, the least squares regression line is:

\[ y = 0.25x + 5.55 \]

This equation is not present in your choices. However, it appears the structure should be checked, as the slope calculated may lead to a different interpretation. If reversing the values leads to a different output that fulfills the requirement, it is worth verifying the dataset's adherence and ensuring the proposed equations meet the generated values per your instructions.

From the provided options, it looks like:

  • \( y = 5.55x - 0.25 \) is the closest to the coefficients found.

Thus, the answer is:

y = 0.25x + 5.55 (if confirmed as calculated accurately) or focus on interpreting the alternate output based on your question's point.