Given that x= 3.5000, sx= 2.2583, y= 4.25000, sy=1.8360, and r= -0.9430, determine the least-squares regression line

To determine the least-squares regression line, we need to find the line that best fits the given data points by minimizing the sum of the squared differences between the observed values and the values predicted by the line.

Given:
- \( \bar{x} = 3.5000 \) (mean of \( x \))
- \( s_x = 2.2583 \) (standard deviation of \( x \))
- \( \bar{y} = 4.2500 \) (mean of \( y \))
- \( s_y = 1.8360 \) (standard deviation of \( y \))
- \( r = -0.9430 \) (correlation coefficient)

### Formula for the Least-Squares Regression Line

The equation of the least-squares regression line \( y = a + bx \) can be determined using the following formulas:

1. **Slope (b)**: 

\[
b = r \cdot \frac{s_y}{s_x}
\]

2. **Y-intercept (a)**: 

\[
a = \bar{y} - b \cdot \bar{x}
\]

### Step-by-Step Calculation

#### Step 1: Calculate the slope \( b \)

Substitute the given values into the formula for the slope:

\[
b = r \cdot \frac{s_y}{s_x}
\]

\[
b = -0.9430 \cdot \frac{1.8360}{2.2583}
\]

Let's compute this:

\[
b = -0.9430 \cdot 0.8128 = -0.7661
\]

#### Step 2: Calculate the y-intercept \( a \)

Substitute the values into the formula for the y-intercept:

\[
a = \bar{y} - b \cdot \bar{x}
\]

\[
a = 4.2500 - (-0.7661) \cdot 3.5000
\]

Let's compute this:

\[
a = 4.2500 + 2.68135 = 6.93135
\]

#### Step 3: Write the least-squares regression line equation

Now, using the calculated slope and intercept, we can write the equation of the regression line:

\[
y = 6.93135 - 0.7661x
\]

### Conclusion

The least-squares regression line is:

\[
y = 6.93135 - 0.7661x
\]

Would you like further details or have any questions?

### Related Questions

1. How does the sign of the correlation coefficient affect the slope of the regression line?
2. What does the slope of the regression line represent in this context?
3. How would you interpret the y-intercept in a real-world scenario?
4. How can we assess the accuracy of the regression model?
5. How does outlier data affect the regression line?

### Tip

When interpreting the slope of the regression line, remember that it represents the change in the dependent variable \( y \) for a one-unit change in the independent variable \( x \).

Learn how to calculate the least-squares regression line using statistics. Given data x = 3.5000, sx = 2.2583, y = 4.2500, sy = 1.8360, and r = -0.9430. Step-by-step solution to find slope and y-intercept. Suitable for advanced high school students and learners interested in statistics and regression analysis.

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