Question

5. Use the data to answer the following questions
x y
2 66
4 105
4 99
4 120
5 100
5 77
5 169
6 200
6 176
(1 point)
Part A: The line of best fit (regression line) is y =

x +
(Round your values to the nearest hundredths).

Part B: Determine the residual data point (2,66)
. (Use the rounded regression line you found above)

Recall that Residual Value = Actual Y-Value - Estimate Y-Value

The Residual Value =
−
=
(Round your values to the nearest hundredths).

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Answer 1

To find the line of best fit (regression line) for the given data, we can use linear regression. The standard form of a linear equation is \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.

Given the data points:

\[
\begin{align*}
(2, 66) \\
(4, 105) \\
(4, 99) \\
(4, 120) \\
(5, 100) \\
(5, 77) \\
(5, 169) \\
(6, 200) \\
(6, 176) \\
\end{align*}
\]

### Part A: Finding the Line of Best Fit

1. **Calculating the necessary sums:**
- \(n\) (number of data points) = 9
- \(\sum x = 2 + 4 + 4 + 4 + 5 + 5 + 5 + 6 + 6 = 45\)
- \(\sum y = 66 + 105 + 99 + 120 + 100 + 77 + 169 + 200 + 176 = 1112\)
- \(\sum xy = 2 \times 66 + 4 \times 105 + 4 \times 99 + 4 \times 120 + 5 \times 100 + 5 \times 77 + 5 \times 169 + 6 \times 200 + 6 \times 176 = 132 + 420 + 396 + 480 + 500 + 385 + 845 + 1200 + 1056 = 4164\)
- \(\sum x^2 = 2^2 + 4^2 + 4^2 + 4^2 + 5^2 + 5^2 + 5^2 + 6^2 + 6^2 = 4 + 16 + 16 + 16 + 25 + 25 + 25 + 36 + 36 = 203\)

2. **Using the formulas for slope \(m\) and intercept \(b\):**
\[
m = \frac{n(\sum xy) - (\sum x)(\sum y)}{n(\sum x^2) - (\sum x)^2}
\]
\[
b = \frac{(\sum y) - m(\sum x)}{n}
\]

3. **Plugging in the values:**
\[
m = \frac{9(4164) - (45)(1112)}{9(203) - (45^2)}
\]
\[
m = \frac{37476 - 49940}{1827 - 2025}
\]
\[
m = \frac{-12464}{-198} \approx 62.92
\]

Now find \(b\):
\[
b = \frac{1112 - 62.92 \cdot 45}{9}
\]
\[
b = \frac{1112 - 2831.4}{9} = \frac{-1719.4}{9} \approx -191.04
\]

4. **Thus, the line of best fit is:**
\[
y = 62.92x - 191.04
\]

### Part B: Calculate the Residual for the Data Point (2, 66)

1. **Estimate Y-Value using the regression line:**
\[
Estimate = 62.92(2) - 191.04 = 125.84 - 191.04 \approx -65.20
\]

2. **Now calculate the residual:**
\[
\text{Residual Value} = \text{Actual Y-Value} - \text{Estimate Y-Value}
\]
\[
\text{Residual Value} = 66 - (-65.20) = 66 + 65.20 = 131.20
\]

### Final Answers:

- **Part A:** The line of best fit is:
\[
y = 62.92x - 191.04
\]

- **Part B:** The residual value for the point (2, 66) is:
\[
\text{Residual Value} = 131.20
\]