Statistics > QUESTIONS & ANSWERS > Dalhousie University - STAT 2060Midterm 2 - Solutions (All)

Dalhousie University - STAT 2060Midterm 2 - Solutions

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DALHOUSIE UNIVERSITY FACULTY OF SCIENCE Department of Mathematics and Statistics STATISTICS 2060 Midterm II Examination SOLUTIONS Date and Time: 9:35-10:55 AM, Monday 15 June, 2009 Name: Student I... D #: The entire exam is worth 50 points. It is closed book, but you may refer to one sheet (8.5 x 11in, both sides) of notes. The exam consists of 6 problems, the number of points allocated to each part of a problem is shown in the left hand column. You may use a calculator and straight edge. 1. If X is a normal random variable with mean 25 and standard deviation 15, compute the following probabilities by standardizing. (a) Compute P(15 ≤ X < 75).(2) • P(15 ≤ X < 75) = P( 15−25 15 ≤ Z < 75−25 15 ) = P(−0.667 ≤ Z < 3.333) • P(15 ≤ X < 75) = Φ(3.33) − Φ(−0.67) = 0.9996 − 0.2514 = 0.7482 (b) Compute P(−15 ≤ X < −5).(2) • P(−15 ≤ X < −5) = P( −15−25 15 ≤ Z < −5−25 15 ) = P(−2.67 ≤ Z < −2.00) • P(−15 ≤ X < −5) = Φ(−2.00) − Φ(−2.67) = 0.0228 − 0.0038 = 0.019 2. For Z, the standard normal random variable with mean 0 and standard deviation 1, compute the following: (a) Compute the value of c, where P(0 ≤ Z ≤ c) = 0.352.(2) • P(0 ≤ X ≤ c) = Φ(c) − Φ(0) = 0.352 • Φ(c) = 0.352 + Φ(0) = 0.352 + 0.500 = 0.852 • c = 1.045 (b) Compute the value of c, for P(c ≤ |Z|) = 0.369.(2) • P(|Z| ≥ c) = 1 − P(|Z| ≤ c) = 1 − P(−c ≤ Z ≤ c) = 0.369 • P(|Z| ≥ c) = 1 − (Φ(c) − (1 − Φ(c))) = 1 − (2Φ(c) − 1) = 0.369 • 2(1 − Φ(c)) = 0.369 • Φ(c) = 1 − 0.369 2 = 0.8155 • c = 0.90 3. During summer months, the weekly demand for Chapman’s ice cream (in 100’s of litres) at the Mumford Road Sobey’s store is a continuous random variable, X, whose probability density function is given as: f(x) = ( 1 3 ( x 3 2 − 1) 1.00 ≤ x ≤ 2.46 0 Otherwise (3) (a) Determine the functional form of the cumulative distribution function, F(x), for rv X. • F(x) = R x −∞ 1 3 ( y 3 2 − 1)dy = 1 3 h y 4 8 | x 1.00 − y| x 1.00i This study source was downloaded by 100000784424693 from CourseHero.com on 04-17-2021 11:25:20 GM [Show More]

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