Height (ft) Interval after (min)
96 68
111 80
76 66
91 72
66 58
108 79
116 84
91 79
The data show the time intervals after an eruption (to the next eruption) of a certain geyser. Find the regression equation, letting the first variable be the independent (x) variable. Find the best predicted tie of the interval after an eruption given that the current eruption has a height of 84 feet. Use a significant level of 0.05.
A. What is the regression equation? Ŷ = __ + ___x
B. What is the best predicted value? Ŷ ͌ __ minutes
To find the regression equation and the best predicted value, you can use a statistical software or a graphing calculator. However, let me guide you through the steps using simple calculations:
Step 1: Calculate the mean (average) of the independent variable (height) and the dependent variable (interval after).
Mean height (x̄) = (96 + 111 + 76 + 91 + 66 + 108 + 116 + 91) / 8 = 94.25 feet
Mean interval after (ȳ) = (68 + 80 + 66 + 72 + 58 + 79 + 84 + 79) / 8 = 73.5 minutes
Step 2: Calculate the deviations from the mean for both variables (height and interval after).
Deviation from mean height (xi - x̄) = [96-94.25, 111-94.25, 76-94.25, 91-94.25, 66-94.25, 108-94.25, 116-94.25, 91-94.25] = [1.75, 16.75, -18.25, -3.25, -28.25, 13.75, 21.75, -3.25]
Deviation from mean interval (yi - ȳ) = [68-73.5, 80-73.5, 66-73.5, 72-73.5, 58-73.5, 79-73.5, 84-73.5, 79-73.5] = [-5.5, 6.5, -7.5, -1.5, -15.5, 5.5, 10.5, 5.5]
Step 3: Calculate the product of the deviations from the mean.
Product of deviations (xi - x̄)*(yi - ȳ) = [1.75 * -5.5, 16.75 * 6.5, -18.25 * -7.5, -3.25 * -1.5, -28.25 * -15.5, 13.75 * 5.5, 21.75 * 10.5, -3.25 * 5.5] = [-9.625, 108.875, 136.875, 4.875, -437.375, 75.625, 228.375, -17.875]
Step 4: Calculate the sum of the squared deviations of the independent variable.
Sum of squared deviations of height (xi - x̄)^2 = (1.75^2 + 16.75^2 + (-18.25)^2 + (-3.25)^2 + (-28.25)^2 + 13.75^2 + 21.75^2 + (-3.25)^2) = 2540.25
Step 5: Calculate the regression coefficients.
Regression coefficient b = Σ[(xi - x̄)*(yi - ȳ)] / Σ(xi - x̄)^2 = (-9.625 + 108.875 + 136.875 + 4.875 - 437.375 + 75.625 + 228.375 - 17.875) / 2540.25 ≈ -0.213
Step 6: Calculate the intercept of the regression line.
Intercept a = ȳ - b*x̄ = 73.5 - (-0.213 * 94.25) = 73.5 + 20.08025 ≈ 93.58025
Therefore, the regression equation is Ŷ = 93.58025 - 0.213x.
Step 7: Calculate the best predicted value for a height of 84 feet.
Ŷ = 93.58025 - 0.213 * 84 ≈ 93.58025 - 17.892 ≈ 75.68825
The best predicted value for an eruption with a height of 84 feet is approximately 75.69 minutes.