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Normal Distribution Exercises For use of the JMP calculator to compute the standard normal distribution and standard normal quantiles, read Sall and Lehman (2001) Chapter 3. In JMP IN Version 4, 5, or 6 software, see Help, Contents, User's Guide, Using the Formula Editor. But use of the JMP calculator is not required or necessary. All of this can be done using a standard normal table. 1. Standard Normal Distribution, Use a standard normal table or the JMP calculator PROBABILITY function NORMAL DISTRIBUTION to "compute" the probability that a standard normal random variable is a. less than -3, 0.0013 b. less than -2, , 0.0228 c. between -3 and -2, , 0.0214 d. less than -1, , 0.1587 e. between -2 and -1, , 0.1359 f. less than 0, , 0.5000 g. between -1 and 0, , 0.3413 h. between 0 and 1, , 0.3413 i. between 1 and 2, , 0.1359 j. between 2 and 3, , 0.0214 k. greater than 3. , 0.0013 2. If you haven't already done so, draw a standard normal curve and divide the area under the curve into eight (8) regions with vertical lines through the Z axis at -3, -2, ..., 3. In each region write the probability of each interval: {Z < -3}, {-3 < Z < -2}, {-2 < Z < -1}, . . ., {2 < Z < 3}, {Z > 3}, which you've already computed. This can be done in JMP calculator by using the PROBABILITY function NORMAL DENSITY. 3. Now use the picture you've just drawn to compute the probability that a standard normal random variable is a. between -1 and +1, b. within two (2) standard deviations of 0, c. within three (3) standard deviations of the mean, d. not between -1 and +1, e. not within two (2) standard deviations of 0, f. not within three (3) standard deviations of the mean. 4. Suppose that Z is a standard normal random variable. a. Using a standard normal table, or the JMP PROBABILITY function NORMAL QUANTILE to find the 1-st percentile, i.e., the number z .01, that satisfies the equation P{Z < z .01} = .01 b. Find the 5-th percentile, z .05. c. Find the 10-th percentile, z .10. d. Find the 90-th percentile, z.90. e. Find the 95-th percentile, z.95. f. Find the 99-th percentile, z .99. 5. Assume that Z is standard normal. a. Find a number z such that P{-z < Z < z} = .90. b. For that number z compute: P{Z < -z} = c. For that number z compute: P{Z < z} = d. For that number z compute: P{Z > z} = e. The number z is which percentile of the standard normal distribution? 6. Assume that Z is standard normal. a. Find a number z such that P{-z < Z < z} = .95. b. Compute: P{Z < -z} = c. Compute: P{Z < z} = d. Compute: P{Z > z} = e. The number z is which percentile of the standard normal distribution? 7. The Normal Family. The yearly growth of dwarf-apple-tree seedlings can be measured by the increase in the length of the central leader. Suppose that the second-year growth of such trees is normally distributed with a mean of 20 cm and a standard deviation of 6 cm. a. Compute the probability that the second-year growth of a randomly selected two-year-old dwarf-apple-tree seedling is less than 15 cm. b. Compute the percentage of such dwarf-apple-tree seedlings that would grow more than 25 cm. c. Compute the fraction of such dwarf-apple-tree seedlings that would be expected to have a second-year growth of between 10 and 30 cm. d. Find a number x such that the second-year growth of 90% of the seedlings is more than x. e. Find two numbers a and b such that the second-year growth of 90% of the seedlings is between a and b and such that the second-year growth of 5% is less than a and the second-year growth of 5% is more than b. 8. A veterinarian found that the average time it takes residents to perform a certain procedure is 12 minutes. Assume that the time it takes residents to perform the procedure is normally distributed with a mean of 12 minutes and a standard deviation of 2 minutes. a. Compute the fraction of residents that you would expect to perform the procedure within two minutes b. c. d. e. f. g. h. of the expected time of 12 minutes. Compute the proportion of residents that you would expect to take less than 10 minutes or more than 14 minutes to perform the procedure. Compute the percentage of residents that you would expect to take between 8 and 10 minutes to perform the procedure. Compute the probability that a randomly selected resident would take between 9 and 11 minutes to perform the procedure. Compute the proportion of residents that you would expect to take between 10 and 12 minutes to perform the procedure. Compute the probability that a randomly selected resident would take more than 15 minutes to complete the procedure. For purposes of planning and scheduling, find the time within which 99% of residents would be expected to complete the procedure. If 50 residents were randomly selected, how many would you expect to be able to perform the procedure in 8 minutes or less? Golde I. Holtzman, Department of Statistics, College of Arts and Sciences, Virginia Tech (VPI) Last updated: September 12, 2006 © Golde I. Holtzman, all rights reserved. URL: http://courseware.vt.edu/users/holtzman/STAT5605/prob03.html