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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