ASSIGNMENT Chapter 4 Statistical Methods NAME M S 184 4 4 6 points Type of Random Variable Identify the following random variables as discrete or continuous a The amount of flu vaccine in a syringe b The heart rate number of beats per minute of an American male c The time it takes a student to complete an examination d The barometric pressure at a given location e The number of registered voters who vote in a national election f You score on the SAT ANSWERS a The amount of flu vaccine in a syringe is measured on an interval so this is a continuous random variable b The heart rate number of beats per minute of an American male is countable starting at whatever number of beats per minute is necessary for survival up to the maximum of which the heart is capable That is if m is the minimum number of beats necessary for survival x can take on the values m m 1 m 2 and is a discrete random variable c The time necessary to complete an exam is continuous as it can take on any value 0 x L where L limit imposed by instructor if any d Barometric atmospheric pressure can take on any value within physical constraints so it is a continuous random variable e The number of registered voters who vote in a national election is countable and is therefore discrete f An SAT score can take on only a countable number of outcomes so it is discrete M S 187 188 223 4 15 3 points A discrete random variable x can assume five possible values 2 3 5 8 and 10 Its probability distribution is shown here x p x 2 15 a What is p 5 3 10 5 8 25 10 25 b What is the probability that x equals 2 or 10 c What is P x 8 ANSWERS a 25 b 40 c 75 4 22 2 points controlling the water hyacinth Entomologists are continually searching for new biological agents to control the water hyacinth one of the world s worst aquatic weeds an insect that naturally feeds on the water hyacinth is the delphacid Female delphacids lay anywhere form one to four eggs onto a water hyacinth blade The Annals of the Entomological Society of American Jan 2005 published a study of the life cycle of a South American delphacid species The following table gives the percentages of water hyacinth blades that have one two three and four delphacod eggs Percentage of Blades One Egg Two Eggs Three eggs 40 54 2 Four eggs 4 Source Sosa A J et al Life history of Megamelus scutellaris with decription of immature stages Annals of the Entomological Society of America Vol 98 No 1 2005 adaptes from Table 1 a One of the water hyacinth blades in the study is randomly selected and x the number of delphacid eggs on the blade is observed Give the probability distribution of x b What is the probability that the blade has at least three delphacid eggs ANSWERS QuickTime and a TIFF Uncompressed decompressor are needed to see this picture 4 134 5 points Reliability of a bridge network In the journal Networks May 1995 a team of Chinese university professors investigated the reliability of several capacitatesflow networks One network examined in the article and illustrated here is a bridge network with arcs a1 a2 a3 a4 a5 and a6 The probability distribution of the capacity x for each of the six arcs is provided in the following table Arc a1 a2 a3 Capacity x 3 2 1 0 2 1 0 1 0 p x Arc Capacity x p x 60 25 10 05 60 30 10 90 10 a4 1 0 90 10 a5 1 0 90 10 a6 2 1 0 70 25 05 Source Lin J et al On reliability evaluation of capacitated flow network in terms of minimal pathsets Networks Vol 25 No 3 May 1995 p 135 Table 1 a1 a2 Source node a3 Sink node a4 a5 a6 a Verify that the properties of a discrete probability distribution are satisfied for each are capacity distribution b Find the probability that the capacity for arc a1 will exceed 1 c Repeat part b for each of the remaining five arcs d Compute and for each arc and interpret the results e One path from the source node to the sink node is through arcs a1 and a2 Find the probability that the system maintains a capacity of more than 1 through the a1 a2 path Assume that the are capacities are independent f Another path from the source node to the sink node is through a1 a3 and a6 Find the probability that the system maintains a capacity of 1 through the a1 a3 a6 Path ANSWERS QuickTime and a TIFF Uncompressed decompressor are needed to see this picture QuickTime and a TIFF Uncompressed decompressor are needed to see this picture QuickTime and a TIFF Uncompressed decompressor are needed to see this picture M S 195 196 4 37 3 points Consider the probability distributions shown here x p x 0 1 2 y 0 1 2 3 4 3 p y 1 8 1 a Use your intuition to find the mean for each distribution How did you arrive at your choice b Which distribution appears to be more variable Why c Calculate and 2 for each distribution Compare these answers with your answers in part a and part b ANSWERS a x 1 y 1 b distribution of x c x 1 2 6 y 1 2 2 4 46 11 points The Showcase Showdown On the popular television game show The Price is Right contestants can play The Showcase Showdown The game involves a large wheel with 20 nickel values 5 10 15 20 95 100 marked on it Contestants spin the wheel once or twice with the objective of obtaining the highest total score without going over a dollar 100 According to the American Statistician Aug 1995 the optimal strategy for the first spinner in a three player game is to spin a second time only if the value of the initial spin is 65 or less Let x represent the score of a single contestant playing The Showcase Showdown Assume a fair wheel i e a wheel with equally likely outcomes If the total of the player s spins exceeds 100 the total is set to 0 a If the player is permitted only one spin of the wheel find the probability distribution for x b Refer to part a find E x and interpret this value c Refer to part a give a range of values within which x is likely to fall d Suppose the play will spin the wheel twice no matter what the outcome of the first spin Find the probability distribution for x e What assumption sis you make to obtain the probability distribution in part d Is it a reasonable assumption f Find and for the probability distribution of part d and interpret the results g Refer to part d what is the …