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 minus 0.25

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

Answer 1

To calculate the least squares regression line from the given data points, we first need to organize the information provided in the table. We have the following points based on the values of \( x \) and \( y \):

\[
\begin{align*}
(-5, 3) \\
(-4, 4) \\
(-3, 7) \\
(-2, 7.5) \\
(-1, 2.5) \\
\end{align*}
\]

Next, we'll typically use a spreadsheet program to calculate the least squares regression line, but since we're doing this by hand conceptually, the formula for the least squares regression line is given by:

\[
y = mx + b
\]

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

1. **Calculate the means of x and y**:
\[
\text{Mean of } x = \frac{-5 + -4 + -3 + -2 + -1}{5} = -3
\]
\[
\text{Mean of } y = \frac{3 + 4 + 7 + 7.5 + 2.5}{5} = 4
\]

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

Where \( n \) is the number of points, \( \Sigma xy \) is the sum of the product of each \( x \) and \( y \), \( \Sigma x \) is the sum of \( x \) values, \( \Sigma y \) is the sum of \( y \) values, and \( \Sigma x^2 \) is the sum of the squares of \( x \).

3. **Compute the necessary sums**:
- \( \Sigma x = -5 + -4 + -3 + -2 + -1 = -15 \)
- \( \Sigma y = 3 + 4 + 7 + 7.5 + 2.5 = 24 \)
- \( \Sigma xy = (-5)(3) + (-4)(4) + (-3)(7) + (-2)(7.5) + (-1)(2.5) = -15 - 16 - 21 - 15 + -2.5 = -69.5 \)
- \( \Sigma x^2 = (-5)^2 + (-4)^2 + (-3)^2 + (-2)^2 + (-1)^2 = 25 + 16 + 9 + 4 + 1 = 55 \)

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

5. **Find \( b \)** (using the point-slope form):
\[
b = \text{mean of } y - m \cdot \text{mean of } x = 4 - 0.25(-3) = 4 + 0.75 = 4.75
\]

6. **Thus, the least squares regression line is**:
\[
y = 0.25x + 4.75
\]

Since the approximate value of \( b \) does not match exactly with the given responses, based on the closest matches, we rely on the correct slope and y-intercept.

Given the responses you've shared, the correct least squares regression line is:

**y = 0.25x + 4.75** (Closest match to the options).

None of the provided answer options exactly match the calculated regression line. However, if rounding it slightly modifies would be acceptable.