Upper level probability #1 - Questions and Answers

Question 1 Events 𝐴 A and 𝐵 B are mutually exclusive. Which of the following statements is also true? a. 𝑃 ( 𝐴 ∩ 𝐵 ) = 𝑃 ( 𝐴 ) + 𝑃 ( 𝐵 ) P(A∩B)=P(A)+P(B) b. 𝑃 ( 𝐴 ∪ 𝐵 ) = 𝑃 ( 𝐴 ) + 𝑃 ( 𝐵 ) P(A∪B)=P(A)+P(B) c. 𝐴 A and 𝐵 B are also independent. d. 𝑃 ( 𝐴 ∩ 𝐵 ) = 𝑃 ( 𝐴 ) 𝑃 ( 𝐵 ) P(A∩B)=P(A)P(B) Answer 1 Events 𝐴 A and 𝐵 B cannot both occur at the same time. That implies: 𝑃 ( 𝐴 ∩ 𝐵 ) = 0. P(A∩B)=0. Using the addition rule: 𝑃 ( 𝐴 ∪ 𝐵 ) = 𝑃 ( 𝐴 ) + 𝑃 ( 𝐵 ) − 𝑃 ( 𝐴 ∩ 𝐵 ) . P(A∪B)=P(A)+P(B)−P(A∩B). Since 𝑃 ( 𝐴 ∩ 𝐵 ) = 0 P(A∩B)=0: 𝑃 ( 𝐴 ∪ 𝐵 ) = 𝑃 ( 𝐴 ) + 𝑃 ( 𝐵 ) . P(A∪B)=P(A)+P(B). Correct choice is 𝑏 b. Answer 2 𝑃 ( 𝐴 ∪ 𝐵 ) = 𝑃 ( 𝐴 ) + 𝑃 ( 𝐵 ) . P(A∪B)=P(A)+P(B). Answer 3 We know 𝑃 ( 𝐴 ∪ 𝐵 ) = 𝑃 ( 𝐴 ) + 𝑃 ( 𝐵 ) − 𝑃 ( 𝐴 ∩ 𝐵 ) P(A∪B)=P(A)+P(B)−P(A∩B), and since 𝐴 A and 𝐵 B are mutually exclusive ( 𝑃 ( 𝐴 ∩ 𝐵 ) = 0 P(A∩B)=0): 𝑃 ( 𝐴 ∪ 𝐵 ) = 𝑃 ( 𝐴 ) + 𝑃 ( 𝐵 ) . P(A∪B)=P(A)+P(B). The correct option is 𝑏 b. Question 2 An analysis of variance of the yields of five different varieties of wheat, observed on one plot each at each of six different locations. The data from this randomized block design are listed here: Varieties Location1 Location2 Location3 Location4 Location5 Location6 A 35.3 31.0 32.7 36.8 37.7 38.1 B 30.7 32.2 35.4 31.7 35.0 32.7 C 33.6 30.1 31.3 35.7 33.5 34.0 D 34.9 35.6 35.2 38.3 35.3 37.6 E 32.4 28.9 29.2 30.7 33.7 32.4 (a) Use the appropriate nonparametric test to determine whether the data provide sufficient evidence to indicate a difference in the yields for the five different varieties of wheat. Test using 𝛼 = 0.05 α=0.05. (b) How do the analysis of variance F test compare with the test in part (a)? Explain. Answer (a) There is sufficient evidence to support the claim that there is a difference in the yields for the five different varieties of wheat. (b) The same conclusion is reached. Question 3 The analysis of variance table for a 3 × 4 3×4 factorial experiment, with factor 𝐴 A at three levels and factor 𝐵 B at four levels, and with two observations per treatment is shown here: Source df SS MS F A 2 5.4 2.65 1.297 B 3 9.1 3.033 1.487 AB 6 4.8 0.8 0.391 Error 12 24.5 Total 23 43.7 (a) Fill in the missing items in the table. (b) Do the data provide sufficient evidence to indicate that factors 𝐴 A and 𝐵 B interact? Test using 𝛼 = 0.05 α=0.05. (c) Do the data provide sufficient evidence to indicate that factors 𝐴 A and 𝐵 B affect the response variable 𝑥 x? Answer (a) Missing values filled. (b) No, there is no sufficient evidence to indicate interaction between factors 𝐴 A and 𝐵 B. (c) No, the data do not provide sufficient evidence to indicate that factors 𝐴 A and 𝐵 B affect the response variable 𝑥 x. Question 4 Select the best answer. Which of the following random variables is geometric? (a) The number of times I have to roll a die to get two 6s. (b) The number of cards I deal from a well-shuffled deck of 52 cards until I get a heart. (c) The number of digits I read in a randomly selected row of the random digits table until I find a 7. (d) The number of 7s in a row of 40 random digits. (e) The number of 6s I get if I roll a die 10 times. Answer (a) Not geometric because it involves getting two successes. (b) Geometric because it involves the number of trials until the first success (getting a heart). (c) Geometric because it involves the number of trials until the first success (finding a 7). (d) Not geometric because the trials are fixed. (e) Not geometric because it involves a fixed number of trials. Question 5 What is Var ( 𝑋 − 𝑌 ) Var(X−Y)? We know that the formula for Var ( 𝑋 + 𝑌 ) Var(X+Y) is Var ( 𝑋 ) + Var ( 𝑌 ) + 2 Cov ( 𝑋 , 𝑌 ) Var(X)+Var(Y)+2Cov(X,Y). Does this mean that for Var ( 𝑋 − 𝑌 ) Var(X−Y) it is just: Var ( 𝑋 ) − Var ( 𝑌 ) − 2 Cov ( 𝑋 , 𝑌 ) ? Var(X)−Var(Y)−2Cov(X,Y)? Answer 1 No. Using the definition of variance: Var ( 𝑋 − 𝑌 ) = Var ( 𝑋 ) + Var ( 𝑌 ) − 2 Cov ( 𝑋 , 𝑌 ) . Var(X−Y)=Var(X)+Var(Y)−2Cov(X,Y). Answer 2 It will be: Var ( 𝑋 ) + Var ( 𝑌 ) − 2 Cov ( 𝑋 , 𝑌 ) . Var(X)+Var(Y)−2Cov(X,Y).