Find the equation of the regression line and predict y when x = 2.5.
x - 1 5 4 2 3 6
y - 25 5 10 20 15 5
http://www.alcula.com/calculators/statistics/linear-regression/
using that regression line calculator
y = -4.29x + 28.3
(2.5, 17.6)
To find the equation of the regression line and predict y when x = 2.5, we need to perform linear regression analysis on the given data points. Here are the step-by-step instructions:
Step 1: Calculate the mean of x (x̄) and y (ȳ).
x̄ = (1 + 5 + 4 + 2 + 3 + 6) / 6 = 3.5
ȳ = (25 + 5 + 10 + 20 + 15 + 5) / 6 = 13.33 (rounded to 2 decimal places)
Step 2: Calculate the deviations from the mean for both x and y.
For each data point, subtract the mean from the individual value:
x - 3.5: -2.5, 1.5, 0.5, -1.5, -0.5, 2.5
y - 13.33: 11.67, -8.33, -3.33, 6.67, 1.67, -8.33
Step 3: Calculate the product of the deviations for each data point (x - 3.5) * (y - 13.33). Round the values to 2 decimal places.
(-2.5 * 11.67) = -29.17
(1.5 * -8.33) = -12.50
(0.5 * -3.33) = -1.67
(-1.5 * 6.67) = -10.00
(-0.5 * 1.67) = -0.83
(2.5 * -8.33) = -20.83
Step 4: Calculate the squared deviations for each x value (x - 3.5)². Round the values to 2 decimal places.
(-2.5)² = 6.25
(1.5)² = 2.25
(0.5)² = 0.25
(-1.5)² = 2.25
(-0.5)² = 0.25
(2.5)² = 6.25
Step 5: Sum all the squared deviations and cross-products.
Sum of (x - 3.5)² = 6.25 + 2.25 + 0.25 + 2.25 + 0.25 + 6.25 = 17.50
Sum of (x - 3.5) * (y - 13.33) = -29.17 - 12.50 - 1.67 - 10.00 - 0.83 - 20.83 = -75.00
Step 6: Calculate the slope (b) of the regression line.
b = Sum of (x - x̄) * (y - ȳ) / Sum of (x - x̄)²
b = -75.00 / 17.50 = -4.29 (rounded to 2 decimal places)
Step 7: Calculate the y-intercept (a) of the regression line.
a = ȳ - b * x̄
a = 13.33 - (-4.29) * 3.5
a = 13.33 + 15.01 = 28.34 (rounded to 2 decimal places)
Step 8: Write the equation of the regression line in the form y = a + b * x.
The equation of the regression line is:
y = 28.34 - 4.29 * x
Step 9: Predict y when x = 2.5 using the regression line equation.
y = 28.34 - 4.29 * 2.5
y = 28.34 - 10.73
y = 17.61 (rounded to 2 decimal places)
Therefore, when x = 2.5, the predicted value of y is approximately 17.61.
To find the equation of the regression line and predict y when x = 2.5, we can use simple linear regression. The regression line is an approximation of the relationship between the independent variable (x) and the dependent variable (y).
1. First, calculate the mean values of x and y.
- Mean of x: (1 + 5 + 4 + 2 + 3 + 6) / 6 = 3.5
- Mean of y: (25 + 5 + 10 + 20 + 15 + 5) / 6 = 13.33 (rounded to two decimal places)
2. Calculate the deviations of x and y from their respective means.
- Deviation of x: x - mean of x
(1-3.5), (5-3.5), (4-3.5), (2-3.5), (3-3.5), (6-3.5) = -2.5, 1.5, 0.5, -1.5, -0.5, 2.5
- Deviation of y: y - mean of y
(25-13.33), (5-13.33), (10-13.33), (20-13.33), (15-13.33), (5-13.33) = 11.67, -8.33, -3.33, 6.67, 1.67, -8.33
3. Calculate the product of the deviations of x and y.
- Product of deviations: deviation of x * deviation of y
(-2.5 * 11.67), (1.5 * -8.33), (0.5 * -3.33), (-1.5 * 6.67), (-0.5 * 1.67), (2.5 * -8.33)
= -29.17, -12.49, -1.67, -10.00, -0.83, -20.8
4. Calculate the squared deviations of x.
- Squared deviations of x: (deviation of x)^2
(-2.5)^2, (1.5)^2, (0.5)^2, (-1.5)^2, (-0.5)^2, (2.5)^2
= 6.25, 2.25, 0.25, 2.25, 0.25, 6.25
5. Calculate the sum of the squared deviations of x.
- Sum of squared deviations of x: sum of squared deviations of x
6.25 + 2.25 + 0.25 + 2.25 + 0.25 + 6.25 = 17.5
6. Calculate the sum of the product of the deviations of x and y.
- Sum of the product of deviations: sum of the product of deviations
-29.17 + -12.49 + -1.67 + -10.00 + -0.83 + -20.8 = -74.96
7. Calculate the slope of the regression line (b).
- slope (b) = sum of the product of deviations / sum of squared deviations of x
b = -74.96 / 17.5 ≈ -4.28 (rounded to two decimal places)
8. Calculate the y-intercept of the regression line (a).
- y-intercept (a) = mean of y - (slope * mean of x)
a = 13.33 - (-4.28 * 3.5) ≈ 27.53 (rounded to two decimal places)
9. Write the equation of the regression line.
- The equation of the regression line is y = a + b * x
y = 27.53 - 4.28x
10. Predict y when x = 2.5 using the regression line equation.
- Substituting x = 2.5 into the equation: y = 27.53 - 4.28 * 2.5
y ≈ 27.53 - 10.70 ≈ 16.83 (rounded to two decimal places)
Therefore, when x = 2.5, the predicted value of y is approximately 16.83.