Question

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

Answer 1

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.