ASSIGNMENT Chapter 5 Statistical Methods NAME M S 231 232 5 5 6 points Suppose x is a random variable best described by a uniform probability distribution with c 2 and d 4 a Find f x ANSWER b Find the mean and standard deviation of x ANSWER c Find P x ANSWER d Find P x 2 78 ANSWER e Find P 2 4 x 3 7 ANSWER f Find P x 2 ANSWER 5 7 3 points The random variable x is best described by a uniform probability distribution with c 100 and d 200 Find the probability that x assume a value a More than two standard deviations from ANSWER b Less than three standard deviations from ANSWER c Within two standard deviations of ANSWER 5 10 3 points Trajectory if an electrical circuit Researchers at the University if California Bekeley have designed built and tested a switched capacitor circuit for generating random signals International Journal of Circuit Theory and Applications May June 1990 The circuit s trajectory was shown to be uniformly distributed on the interval 0 1 a Give the mean and variance of the circuit s trajectory ANSWER b Compute the probability that the trajectory falls between 2 and 4 ANSWER c Would you expect to observe a trajectory that exceeds 995 Explain ANSWER 5 11 4 points Maintaining pipe wall temperature Maintaining a constant temperature in a pipe wall in some hot process applications is critical A new technique that utilizes bolt on trace elements to maintain the temperature was presented in the Journal if Heat Transfer November 2000 Without bolt on trace elements the pipe wall temperature of a switch condenser used to produce plastic has a uniform distribution ranging from 260o to 290oF When several bolt on trace elements are attached to the piping the wall temperature is uniform to 278o to 285oF a Ideally the pipe wall temperature should range between 280o and 284oF What is the probability that the temperature will fall into this ideal range when no bolt on trace elements are used When bolt on trace elements are attached to the pipe ANSWER b When the temperature is 268oF or lower the hot liquid plastic hardens or plates causing a buildup in the piping What is the probability of the plastic plating when mo bolt on trace elements are used When bolt on trace elements are attached to the pipe ANSWER M S 246 248 5 24 8 points Find the following probabilities for the standard normal random variable z a P z 1 46 ANSWER b P z 1 56 ANSWER c P 67 z 2 41 ANSWER d P 1 96 z 33 ANSWER e P z 0 ANSWER f P 2 33 z 1 50 ANSWER g P z 2 33 ANSWER h P z 2 33 ANSWER 5 28 8 points Find a value zo of the standard normal random variable z such that a P z zo 0401 ANSWER b P zo z zo 95 ANSWER c P zo z zo 90 ANSWER d P zo z zo 8740 ANSWER e P zo z 0 2967 ANSWER f P 2 z zo 9710 ANSWER g P z zo 5 ANSWER h P z zo 0057 ANSWER 5 32 4 points Suppose x is a normally distributed random variable with 30 and 8 Find a value xo of the random variable x such that a P x xo 5 ANSWER b P x xo 025 ANSWER c P x xo 10 ANSWER d P x xo 95 ANSWER 5 36 3 points Dental anxiety study Psychology students at Wittenberg University completed the Dental Anxiety Scale questionnaire Psychology Reports Aug 1997 Scores on the scale range from 0 no anxiety to 20 extreme anxiety The mean score was 11 and the standard deviation was 3 5 Assume that the distribution of all scores on the Dental Anxiety Scale is normal with 11 and 3 5 a Suppose you score a 16 on the Dental Anxiety Scale Find the z value for this score ANSWER b Find the probability that someone scores between 10 and 15 on the Dental Anxiety Scale ANSWER c Find the probability that someone scores above 17 on the Dental Anxiety Scale ANSWER 5 44 4 points alcohol threats and electric shocks A group of Florida State University psychologists examined the effects of alcohol on the reactions of people to a threat Journal of Abnormal psychology Vol 107 1998 After obtaining a specified blood alcohol level the psychologists placed experimental subjects in a room and threatened them with electric shocks Using sophisticated equipment to monitor the subjects eye movements the psychologists recorded the startle response measured in milliseconds of each object The mean and standard deviation of the startle responses were 37 9 and 12 4 respectively Assume that the startle response x for a person with the specified blood alcohol level is approximately normally distributed a Find the probability that x is between 40 and 50 milliseconds ANSWER b Find the probability that x is less than 30 milliseconds ANSWER c Given an interval for x centered around 37 9 milliseconds so that the probability that x falls in the interval is 95 ANSWER d Ten percent of the experimental subjects have startle responses above what value ANSWER 4 45 3 points Visually impaired students The Journal of visual impairment Blindness May June 1997 published a study of the lifestyles of visually impaired students Using diaries the students kept track of several variables including number of hours of sleep obtained in a typical day These visually impaired students had a mean of 9 06 hours and a standard deviation of 2 11 hours Assume that the distribution of the number of hours of sleep for this group of students is approximately normal a Find the probability that a visually impaired student obtains less than 6 hours of sleep on a typical day ANSWER b Find the probability that a visually impaired student gets between 8 and 10 hours of sleep on a typical day ANSWER c Twenty percent of all visually impaired students obtain less than how many hours of sleep on a typical day ANSWER M S 246 262 263 272 5 23 4 points Find the area under the standard normal distribution between the following pairs of z scores a z 2 00 and z 0 ANSWER b z 1 00 ANSWER and z 0 c z 1 69 ANSWER and z 0 d z 58 ANSWER and z 0 5 79 4 points Melanoma deaths According to the American Cancer Society melanoma a form of skin cancer kills 15 of Americans who suffer from the disease each year Consider a sample of 10 000 melanoma patients a What are the expected value and variance of x the number of the 10 000 melanoma patients who die of the affliction this year ANSWER b Find the probability that x will exceed 1 600 patients per year ANSWER c Would you expect x the number of patients dying of melanoma to exceed 6 500 in any single year Explain ANSWER 5 88 3 points Waiting time at a doctor …