University:
Boston University
Course:
CGS MA 121 | Calculus
Academic year:

2024
Views:

145

Pages:

5
Author:

fiziokrat16l

Question 1 There are 10 balls in a bag: 4 red, 3 blue, 1 green, and 2 yellow. Answer each question by reporting probability as a simplified fraction, a decimal, or a percent. What is the probability that you will draw yellow first and blue second? Answer p = no of favorable outcomes / no of outcomes p = 3 / 10 probability that blue is drawn given yellow first and second is given below p = 2 / (3 * 9) p = (2*1) / (3 * 10 - 1) p = 6 / 96 p = 1/15 required probability = 1/15 Question 2 The probability is 0.4 that the final exam is difficult. If the final exam is difficult, the probability is 0.3 that Nancy will pass the course. If the final exam is not difficult, the probability is 0.9 that Nancy will pass the course. What is the probability that Nancy will pass the course? Answer D = exam was difficult group S = Nancy will pass the test group P(D) = 0.4 P(D’) = 1 - 0.4 = 0.6 P(S|D) = 0.3 P(S|D’) = 0.9 probability that nancy will pass the exam P(S) = (P(D) * P(S|D)) + (P(D’) * P(S|D’)) = (0.4 * 0.3) + (0.6 * 0.9) = 0.12 + 0.54 = 0.66 0.66 = probability that nancy will pass Question 3 The probability is 0.4 that the final exam is difficult. If the final exam is difficult, the probability is 0.3 that Nancy will pass the course. If the final exam is not difficult, the probability is 0.9 that Nancy will pass the course. Given the observation that Nancy passed the course, what is the probability that the final exam was difficult? [Use Bayes' Theorem.] Answer D = difficult exam S = nancy will pass exam P(D) = 0.4 P(D’) = 1 - 0.4 = 0.6 P(S|D) = 0.3 P(S|D’) = 0.9 probability that the exam was difficult P(D|S) = P(S|D) * P(D) / P(S) where, P(S) = 0.66 P(D|S) = (0.3 * 0.4) / 0.66 = 0.12 / 0.66 = 0.1818 Question 4 Suppose we are testing H₀: p = .20 vs Hₐ: p>.20 and TS=2.34. What is the p-value? Answer Given: (TS=test statistic z) H₀: p = 0.20 Hₐ: p > 0.20 z = 2.34 The P-value is the probability of obtaining the value of the test statistic, or a value more extreme, when the null hypothesis is true. Determine the P-value using the normal probability table in the appendix. P = P(Z > 2.34) = 1 - P(Z < 2.34) = 1 - 0.9904 = 0.0096 Question 5 Find the probabilities for the standard normal random variable z: P(z>1.34) Answer Using the appendix's normal probability table, calculate the corresponding probability. P(z < 1.34) is provided in the appendix's standard normal probability table in the row beginning with 1.3 and the column beginning with .04. P(z < 1.34) = 0.9099 P(A') = 1 - P(A) P(z > 1.34) = 1 - P(z < 1.34) = 1 - 0.9099 = 0.0901