University:
Massachusetts Institute of Technology
Course:
15.075J | Statistical Thinking and Data Analysis
Academic year:

2024
Views:

14

Pages:

5
Author:

Pierre C.

15.075 Exam 1 Grading is based on demonstration of conceptual understanding, so you need to show all of your work. Problem 1 A very large bin contains 3 diﬀerent types of disposable ﬂashlights. The probability that a type 1 ﬂashlight will give over 100 hours of use is 0.7, with the corresponding probabilities for type 2 and 3 ﬂashlights being 0.4 and 0.3 respectively. Suppose that 20% of the ﬂashlights in the bin are type 1, 30% are type 2 and 50% are type 3. 1. What is the probability that a randomly chosen ﬂashlight will give more than 100 hours of use? 2. Given the ﬂashlight lasted over 100 hours, what is the conditional probability that it was a type 1 ﬂashlight? Solution a. Let A be the event {the ﬂashlight will give more than 100 hours of use } and Bi be the event {we choose ﬂashlight of type i}. Then we have: P (A) = 3 3 P (A/Bi )P (Bi ) = 0.7 · 0.2 + 0.4 · 0.3 + 0.3 · 0.5 = 0.41 i=1 b. We have: P (B1 ) = P (B1 /A · P (A) 0.7 · 0.2 = = 0.34 P (A) 0.41 1 Problem 2 In HighHatVille (population 1200), a third of the townsfolk have very high hats, more than three feet tall for the hat (not including the person underneath)! But they keep their hats hidden most of the time and bring them out only on HighHatHoliday. They lend the hats out when they aren’t using them though. The MIT students need some high hats for a party, so we decided to visit HighHatVille for a day. Unfortunately we can’t spend too long there, and we need 50 hats. How many people do we need to visit in order to ensure that we get at least 50 hats with more than 70% probability? Hint: We do not expect a numerical answer just a procedure which would give the exact answer. Solution Let n be the number of people we visit. Then the number of hats we get follows a hypergeometric distribution with N = 1200 and M = 400. We want the probability of getting at least 50 hats to be greater than 0.7, in other words: n 800 n 400 n 3 · n−i i n ≥ 0.7 1200 i=50 n We can solve it with MATLAB and ﬁnd n=158 (we expected a value close but greater to 150). 2 Problem 3 The annual snowfall at a particular city is modeled as a normal random variable with a mean of µ = 60 inches and a standard deviation of σ = 20. What is the probability that this year’s snowfall will be between 40 and 100 inches? Solution Let X the annual snowfall. It is: P (40 ≤ X ≤ 100) = P ( 40 − 60 X − 60 100 − 60 ≤ ≤ ) = P (−1 ≤ Z ≤ 2) = Φ(2) − Φ(−1) = 0.8186. 20 20 20 3 Problem 4 An airline company accepts 100 reservations for a ﬂight that has 72 available seats. A person booking a reservation does not show up in the ﬂight with probability q = 0.1. Find an upper bound in the probability that no traveler showing up is bumped (excluded) from the ﬂight. Solution Let X be the number of people showing up in the ﬂight. Then X is a Bin(n, p) with n = 100 and p = 0.9. It is E[X] = n·p = 100·0.9 = 90 and var(X) = np(1−p) = 9. We are interested in P (X ≤ 72). We have: P (|X − 90| ≥ 18) = P (X ≥ 108) + P (X ≤ 72) = P (X ≤ 72) But from Chebyshev’s inequality we know: P (|X − E[X]| ≥ 18) ≤ var(X) 1 = . 2 18 36 Therefore, P (X ≤ 72) ≤ 4 1 . 36 Problem 5 Provide a sample that could correspond to this box plot: Solution Obviously, there are uncountably inﬁnite solutions to this problem. Two representative samples are: {25, 40, 50, 52, 53, 55, 56, 58, 60, 68, 80} and {25, 40, 48, 50, 51, 52, 53, 55, 56, 57, 58, 60, 65, 68, 80} 5

Statistical Thinking and Data Analysis, Exam 1 Solutions

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