Exam 3 in MATH 3030: Probability and Statistics, Exams of Mathematics

Download Exam 3 in MATH 3030: Probability and Statistics and more Exams Mathematics in PDF only on Docsity! MATH 3030 Exam #3 Name ______________________________ 12 April 2012 Show ALL Work!! 1. Represent the following data by a stem-and-leaf plot or a box plot, your choice. (5 points) 203 199 198 201 200 201 201 19 8 9 20 0 1 1 1 3 198 200 201 203 Note that the median and upper quartile are both 201. 2. Three screws are drawn at random from a lot of 100 screws, 10 of which are defective. Find the probability of each of the following. (5 points each) a. All three screws are defective if drawn without replacement. 10 9 8 720 0.000742 100 99 98 970200 ⎛ ⎞⎛ ⎞⎛ ⎞ = =⎜ ⎟⎜ ⎟⎜ ⎟ ⎝ ⎠⎝ ⎠⎝ ⎠ b. All three screws are defective if drawn with replacement. 310 1 0.001 100 1000 ⎛ ⎞ = =⎜ ⎟ ⎝ ⎠ c. At least 1 screw is defective if drawn with replacement. 1 – P(0 defective) = 390 7291 1 0.271 100 1000 ⎛ ⎞− = − =⎜ ⎟ ⎝ ⎠ 3. Suppose the number of hits to a web site occur at the rate of 1.4 per minute between 7:00 pm and 9:00 pm. (5 points each) a. What is the probability distribution function for this scenario? ( ) 1.41.4( ) ! x ttf x x −= e where t is the length of time under consideration and x is the number of hits a) _______________________ b. What is the probability that the number of hits to the web site between 7:30 and 7:35 is exactly 7? ( )( ) ( )( ) 7 7 1.4 5 71.4 5 7(7) 0.149003 7! 7! f e e− − ⎡ ⎤⎣ ⎦= = = b) _______________ Page 2 c. What is the probability that the number of hits between 7:30 and 7:35 is more than 7? 1 – P(x < 7) = 7 2 3 71 1 11 1 7 7 7 7 1 0.598714 0.401286 2 6 5040 e− ⎛ ⎞− + + + + + = − =⎜ ⎟ ⎝ ⎠ c) _______________ 4. The following table is a probability distribution where the random variable x represents the number of activities at least one parent of a K-5th grade student is involved in. Based on this distribution, what is the expected number of activities a parent is involved in? (5 point) x P(x) 0 0.035 1 0.074 2 0.197 3 0.320 4 0.374 ( )( ) 0 .074 .394 .960 1.496 2.924i iE x x P x= = + + + + =∑ 5. Find the mean of the random variable X with probability function 4( ) 4 xf x e−= x > 0. (10 points) ( ) ( )4 4 4 4 4 4 4 00 0 0 0 1 14 lim 4 lim lim lim 0 4 4 b b bbx x x x b x b b b b b x e dx x e dx xe e dx be e e ∞ − − − − − − − →∞ →∞ →∞ →∞ ⎡ ⎤ ⎡ ⎤⎛ ⎞⎡ ⎤= = − + = − − = − −⎢ ⎥ ⎜ ⎟⎢ ⎥⎢ ⎥⎣ ⎦ ⎝ ⎠⎣ ⎦⎢ ⎥⎣ ⎦ ∫ ∫ ∫ 1 1 4 4 = 6. Compute the probability of obtaining fewer than two “Six” in rolling a fair die 4 times. (5points) Fewer than 2 = P(0) + P(1) = 0 4 1 34 41 5 1 5 .4822 .3858 0.8680 0 16 6 6 6 ⎛ ⎞ ⎛ ⎞⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞+ = + =⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎝ ⎠ ⎝ ⎠ 7. The probability that a regularly scheduled flight departs on time is P(d) = 0.83; the probability that it arrives on time is P(a) = 0.82; and the probability that it departs and arrives on time is 0.78. Find the probability that a plane arrives on time given that it departed on time. (5 points) ( ) 0.78( | ) 0.939759 ( ) 0.83 P arrive departP arrive depart P depart ∩ = = = 8. Let X be normal with mean 105 and variance 25. Find P(X < 112.5). (5 points) P(X < 112.5) = 112.5 105 ( 1.5) 0.9332 5 P z P z−⎛ ⎞≤ = ≤ =⎜ ⎟ ⎝ ⎠