Which equation of the least squares regress matches the data set?

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To determine the **least squares regression equation** that best fits your data set, you'll need to follow these steps. The least squares method finds the line (in the case of linear regression) that minimizes the sum of the squared differences (errors) between the observed values and the values predicted by the line.

### **1. Gather Your Data**
Ensure you have your data organized into pairs of $$ (x, y) $$ values. For example:

| $$ x $$ | $$ y $$ |
|---------|---------|
| 1       | 2       |
| 2       | 3       |
| 3       | 5       |
| 4       | 4       |
| 5       | 6       |

### **2. Calculate Necessary Summations**
Calculate the following for your data set:
- **Sum of $$ x $$:** $$ \sum x $$
- **Sum of $$ y $$:** $$ \sum y $$
- **Sum of $$ xy $$:** $$ \sum xy $$
- **Sum of $$ x^2 $$:** $$ \sum x^2 $$
- **Number of data points:** $$ n $$

Using the example data:

$$
\begin{align*}
\sum x &= 1 + 2 + 3 + 4 + 5 = 15 \
\sum y &= 2 + 3 + 5 + 4 + 6 = 20 \
\sum xy &= (1 \times 2) + (2 \times 3) + (3 \times 5) + (4 \times 4) + (5 \times 6) = 2 + 6 + 15 + 16 + 30 = 69 \
\sum x^2 &= 1^2 + 2^2 + 3^2 + 4^2 + 5^2 = 1 + 4 + 9 + 16 + 25 = 55 \
n &= 5
\end{align*}
$$

### **3. Use the Least Squares Formulas**
The formulas to calculate the slope ($$ m $$) and y-intercept ($$ b $$) of the regression line $$ y = mx + b $$ are:

$$
m = \frac{n(\sum xy) - (\sum x)(\sum y)}{n(\sum x^2) - (\sum x)^2}
$$

$$
b = \frac{(\sum y) - m(\sum x)}{n}
$$

Plugging in the example values:

$$
m = \frac{5(69) - (15)(20)}{5(55) - (15)^2} = \frac{345 - 300}{275 - 225} = \frac{45}{50} = 0.9
$$

$$
b = \frac{20 - 0.9(15)}{5} = \frac{20 - 13.5}{5} = \frac{6.5}{5} = 1.3
$$

### **4. Write the Regression Equation**
Using the calculated slope and y-intercept:

$$
y = 0.9x + 1.3
$$

This equation represents the line of best fit for the provided data set using the least squares method.

### **Additional Considerations**
- **Coefficient of Determination ($$ R^2 $$)**: Indicates how well the regression line fits the data. An $$ R^2 $$ value closer to 1 means a better fit.
- **Residual Analysis**: Analyze the differences between observed and predicted values to assess the fit.

### **Need Specific Assistance?**
If you provide your specific data set or further details, I can help you calculate the exact least squares regression equation tailored to your data.
To find the least squares regression equation for your data set, follow these steps: 1. **Gather Data**: Collect your $$ (x, y) $$ pairs. 2. **Calculate Summations**: Find $$ \sum x $$, $$ \sum y $$, $$ \sum xy $$, $$ \sum x^2 $$, and count the number of data points $$ n $$. 3. **Compute Slope ($$ m $$) and Y-Intercept ($$ b $$)**: $$ m = \frac{n(\sum xy) - (\sum x)(\sum y)}{n(\sum x^2) - (\sum x)^2} $$ $$ b = \frac{(\sum y) - m(\sum x)}{n} $$ 4. **Write the Equation**: Use $$ m $$ and $$ b $$ to form $$ y = mx + b $$. For example, with the data: $$ \begin{align*} \sum x &= 15 \ \sum y &= 20 \ \sum xy &= 69 \ \sum x^2 &= 55 \ n &= 5 \end{align*} $$ Calculate $$ m = 0.9 $$ and $$ b = 1.3 $$, resulting in the equation: $$ y = 0.9x + 1.3 $$ If you provide your specific data, I can help you find the exact regression equation.

Which equation of the least squares regress matches the data set?

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