Math Problem Statement

Compute the least-squares regression line for predicting y from x given the summary statistics: x̄ = 6, sx = 4, ȳ = 103, sy = 103, r = -0.84. Round the slope and y-intercept to at least four decimal places.

Solution

To solve the given problem and compute the least-squares regression line for predicting yy from xx, we use the formula for the regression line:

y^=b0+b1x\hat{y} = b_0 + b_1x

Step 1: Compute the slope (b1b_1)

The formula for the slope is:

b1=rsysxb_1 = r \cdot \frac{s_y}{s_x}

Where:

  • r=0.84r = -0.84 (correlation coefficient),
  • sy=103s_y = 103 (standard deviation of yy),
  • sx=4s_x = 4 (standard deviation of xx).

Substituting the values:

b1=0.841034=0.8425.75=21.63b_1 = -0.84 \cdot \frac{103}{4} = -0.84 \cdot 25.75 = -21.63

Step 2: Compute the y-intercept (b0b_0)

The formula for the y-intercept is:

b0=yˉb1xˉb_0 = \bar{y} - b_1\bar{x}

Where:

  • xˉ=6\bar{x} = 6 (mean of xx),
  • yˉ=103\bar{y} = 103 (mean of yy),
  • b1=21.63b_1 = -21.63 (slope).

Substituting the values:

b0=103(21.63)(6)b_0 = 103 - (-21.63)(6) b0=103+129.78=232.78b_0 = 103 + 129.78 = 232.78

Final Regression Line Equation:

y^=232.780021.6300x\hat{y} = 232.7800 - 21.6300x

Would you like further details or clarification?


Related Questions:

  1. How does the correlation coefficient rr influence the slope of the regression line?
  2. What is the significance of the standard deviations sxs_x and sys_y in this context?
  3. How do you interpret the y-intercept in the context of this regression?
  4. Why is rounding to four decimal places important in regression analysis?
  5. Can the regression line be used for predictions outside the observed range of xx? Why or why not?

Tip:

Always double-check the computation of b1b_1 and b0b_0, as even small rounding errors can affect the regression line's accuracy.

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Math Problem Analysis

Mathematical Concepts

Statistics
Regression Analysis
Correlation

Formulas

b1 = r * (sy / sx)
b0 = ȳ - b1 * x̄
ŷ = b0 + b1 * x

Theorems

Least-Squares Regression

Suitable Grade Level

Grades 11-12, College Level