Regression Equation and Probability Addition Theorem Solutions

14. Obtain the regression equation of Y onX and correlation coefficient for the following: X; 4 6 8 10 12 , f ;7 9 8 12 15 1. Calculate the necessary sums: X Y X² XY 2 10 4 20 3 9 9 27 7 11 49 77 8 8 64 64 10 12 100 120 ΣX = 30 ΣY = 50 ΣX² = 226 ΣXY = 308 Export to Sheets 2. Calculate the slope (b): b = (nΣXY - ΣXΣY) / (nΣX² - (ΣX)²) where n is the number of data points (n = 5 in this case) b = (5 * 308 - 30 * 50) / (5 * 226 - 30²) b = (1540 - 1500) / (1130 - 900) b = 40 / 230 b ≈ 0.174 3. Calculate the y-intercept (a): a = (ΣY - bΣX) / n a = (50 - 0.174 * 30) / 5 a = (50 - 5.22) / 5 a ≈ 8.956 4. The fitted line: Substitute the values of a and b into the equation Y = a + bX: Y = 8.956 + 0.174X Therefore, the fitted straight line is Y = 8.956 + 0.174X

15. State and prove addition theorem for two events. He addition theorem of probability calculates the probability of the union of two events. It states that the probability of either of the two events happening is equal to the sum of the individual probabilities of the events happening, minus the probability of both events happening. Statement: For any two events A and B, the probability of their union is given by: P(A ∪ B) = P(A) + P(B) - P(A ∩ B) where:  P(A ∪ B) represents the probability of event A or event B occurring.  P(A) represents the probability of event A occurring.  P(B) represents the probability of event B occurring.  P(A 1 ∩ B) represents the probability of both events A and B occurring. 2 The proof of the addition theorem can be visualized using a Venn diagram. Let's break down the Venn diagram:  The entire sample space (S) represents all possible outcomes.  Circle A represents the outcomes where event A occurs.  Circle B represents the outcomes where event B occurs.  The overlapping region represents the outcomes where both events A and B occur (A ∩ B). From the Venn diagram, we can observe:  P(A) = P(A - B) + P(A ∩ B)  P(B) = P(B - A) + P(A ∩ B)  P(A ∪ B) = P(A - B) + P(B - A) + P(A ∩ B) Substituting the first two equations into the third equation, we get: P(A ∪ B) = [P(A) - P(A ∩ B)] + [P(B) - P(A ∩ B)] + P(A ∩ B) Simplifying the equation:P(A ∪ B) = P(A) + P(B) - P(A ∩ B)This proves the addition theorem of probability.