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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=0.25x−5.55
𝑦
=
0.25
𝑥
−
5.55
y equals 0.25 x minus 5.55

y=0.25x+5.55
𝑦
=
0.25
𝑥
+
5.55
y equals 0.25 x plus 5.55

y=5.55x−0.25
𝑦
=
5.55
𝑥
−
0.25
y equals 5.55 x minus 0.25

y=5.55x+0.25
𝑦
=
5.55
𝑥
+
0.25

1 answer

To determine the least squares regression line from the given data points, we first need to calculate the values necessary for the regression equation of the form \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.

Given the data points:

| x | y | |------|------| | -5 | -5 | | -4 | -4 | | 3 | -3 | | 4 | -3 | | 7 | -2 | | 7.5 | -1 | | 2.5 | -1 |

Now we will find \( m \) (slope) and \( b \) (y-intercept) using the formulas:

  1. \( m = \frac{N(\sum xy) - (\sum x)(\sum y)}{N(\sum x^2) - (\sum x)^2} \)
  2. \( b = \frac{\sum y - m(\sum x)}{N} \)

Where:

  • \( N \) is the number of data points,
  • \( \sum x \) is the sum of x-values,
  • \( \sum y \) is the sum of y-values,
  • \( \sum xy \) is the sum of the product of x and y-values,
  • \( \sum x^2 \) is the sum of the squares of the x-values.

Let's compute these sums:

  1. Count the number of data points \( N = 7 \).
  2. Sum of x-values \( \sum x = -5 - 4 + 3 + 4 + 7 + 7.5 + 2.5 = 15 \).
  3. Sum of y-values \( \sum y = -5 - 4 - 3 - 3 - 2 - 1 - 1 = -19 \).
  4. Sum of products \( \sum xy = (-5)(-5) + (-4)(-4) + (3)(-3) + (4)(-3) + (7)(-2) + (7.5)(-1) + (2.5)(-1) \) = \( 25 + 16 - 9 - 12 - 14 - 7.5 - 2.5 = -4 \).
  5. Sum of squares \( \sum x^2 = (-5)^2 + (-4)^2 + (3)^2 + (4)^2 + (7)^2 + (7.5)^2 + (2.5)^2 \) = \( 25 + 16 + 9 + 16 + 49 + 56.25 + 6.25 = 122.5 \).

Now using the formulas:

  1. Calculate the slope \( m \):

    \[ m = \frac{7(-4) - (15)(-19)}{7(122.5) - (15)^2} = \frac{-28 + 285}{857.5 - 225} = \frac{257}{632.5} \approx 0.406 \]

  2. Calculate the y-intercept \( b \):

    \[ b = \frac{-19 - 0.406(15)}{7} = \frac{-19 - 6.09}{7} \approx \frac{-25.09}{7} \approx -3.58 \]

Putting it together, the least squares regression line is approximately:

\[ y = 0.406x - 3.58 \]

It appears that I cannot match exactly to any of the provided options without additional rounding or calculation errors.

Upon reviewing the list of responses, none of the computed values for \( m \) and \( b \) directly match the given choices:

  1. \( y = 0.25x - 5.55 \)
  2. \( y = 0.25x + 5.55 \)
  3. \( y = 5.55x - 0.25 \)
  4. \( y = 5.55x + 0.25 \)

Thus, further verification of calculations or additional context regarding rounding or method could be necessary to reconcile this discrepancy.