# A calculate the probability that a randomly selected

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(a) Calculate the probability that a randomly selected student has height between 5 . 45 and 5 . 85 feet. (b) What is the proportion of students above 6 feet? Problem 5.5 The raw scores in a national aptitude test are normally distributed with mean 506 and standard deviation 81. (a) What proportion of the candidates scored below 574? (b) Find the 30th percentile of the scores. Problem 5.6 Scores on a certain nationwide college entrance examination follow a normal distribution with a mean of 500 and a standard deviation of 100. (a) If a school admits only students who scores over 670, what proportion of the student pool will be eligible for admission? (b) What admission requirements would you see if only the top 15% are to be eligible? Problem 5.7 A machine is designed to cut boards at a desired length of 8 feet. However, the actual length of the boards is a normal random variable with standard deviation 0 . 2 feet. The mean can be set by the machine operator. At what mean length should the machine be set so that only 5 per cent of the boards are under cut (that is, under 8 feet)? Problem 5.8 The temperature reading X from a thermocouple placed in a constant- temperature medium is normally distributed with mean μ , the actual temperature of the medium, and standard deviation σ . (a) What would the value of σ have to be to ensure that 95% of all readings are within 0 . 1
5.3. EXERCISES 99 of μ ? (b) Consider the difference between two observations X 1 and X 2 (here we could assume that X 1 and X 2 are i.i.d. ), what is the probability that the absolute value of this difference is at most 0 . 075 ? Problem 5.9 Suppose the random variable X follows a normal distribution with mean μ = 50 and standard deviation σ = 5. (a) Calculate the probability P ( | X | > 60). (b) Calculate E X 2 and the interquartile range of X . 5.3.2 Exercise Set B Problem 5.10 Let Z be a standard normal random variable. Find: (a) P ( Z < 1 . 3) (b) P (0 . 8 < Z < 1 . 3) (c) P ( - 0 . 8 < Z < 1 . 3) (d) P ( - 1 . 3 < Z < - 0 . 8) (e) c such that P ( Z < c ) = 0 . 9032 (f) c such that P ( Z < c ) = 0 . 0968 (g) c such that P ( - c < Z < c ) = 0 . 90 (h) c such that P ( | Z | < c ) = 0 . 95 (i) c such that P ( | Z | > c ) = 0 . 80 Problem 5.11 Let X be a normal random variable with mean 10 and variance 25 Find: (a) P ( X < 13) (b) P (11 < X < 13) (c) P (8 < X < 13) (d) P (6 < X < 8) (e) c such that P ( X < c ) = 0 . 9032 (f) c such that P ( X < c ) = 0 . 0968 (g) c such that P ( - c < X - 10 < c ) = 0 . 90 (h) c such that P ( - c < X < c ) = 0 . 95 (j) c such that P ( | X - 10 | > c ) = 0 . 80 (k) c such that P ( | X | > c ) = 0 . 80
100 CHAPTER 5. NORMAL DISTRIBUTION Problem 5.12 A scholarship is offered to students who graduate in the top 5% of their class. Rank in the class is based on GPA (4.00 being perfect). A professor tells you the marks are distributed normally with mean 2.64 and variance 0.5831. What GPA must you