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Sampling Theory Exercises
Use a Standard Normal CDF Table or the JMP calculator PROBABILITY function
NORMAL DISTRIBUTION or Minitab Calc and Graph menus to "compute" the
standard normal probabilities. In fact, the entire exercise can be done on JMP or Minitab,
but doing it by hand is just a good, and perhaps better.
1. Dwarf Apple Trees (This builds on the earlier exercise). 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 fraction of such dwarf-apple-tree seedlings that would be
expected to have a second-year growth of between 18 and 22 cm. [Note:
The fraction, i.e., the proportion, of a population (in this case, the
population is all such dwarf-apple-tree seedlings) that would be expected
to have a certain property (in this case, second-year growth of between 18
and 22 cm) is the same thing as the probability that a randomly selected
individual (in this case, a randomly selected dwarf apple tree) has that
property. I.e., the expected population proportion and the probability are
the same thing. Moreover, the probability that a, i.e., one, randomly
selected individual has a certain property is, technically, the probability
that the mean of a sample of size n = 1 has that property. This is because
the mean of a sample of size one is simply the value of the one
observation.]
b. Now consider the sampling distribution of the sample mean for samples of
size 4. Compute the expected value , i.e., mean, of the sampling
distribution.
c. Compute the standard deviation of the sampling distribution of the mean
for samples of size 4.
d. For samples of size 5, compute the probability that the sample mean is
between 18 and 22 cm.
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e. For samples of size 25, compute the probability that the sample mean is
between 18 and 22 cm.
f. For samples of size 100, compute the probability that the sample mean is
between 18 and 22 cm.
g. Graph the probability that the mean is between 18 and 22 cm as a function
of the sample size for sample sizes of 1 to 100. [Note: You calculated
P {18 < Y < 22} for n = 1 in part (a) above, for n = 5 in part (d) above, for
n = 25 in part (e) above, and for n = 100 in part (f) above. Thus, you have
four points to plot on the graph. You can simply draw this by hand and
connect them with a smooth curve. Or you can calculate
P {18 < Y < 22} for additional values of n if you want to get a more precise
graph. Or you can calculate it for all integer values of n between 1 and 100
in JMP or Minitab, and then use the software to draw a very precise graph.
But drawing the graph by hand with just the 4 points will be sufficient for
you to get the idea, which is all that’s important. ]
2. 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 probability that a randomly selected resident would take
between 11 and 13 minutes to perform the procedure, i.e., within 1.0
minute of the mean.
b. Graph the probability that the sample mean would be between 11 and 13
minutes, for samples of size 1 to 100. [Note: It will be sufficient to do this
the same as explained in Exercise (1.g) above.]
c. If you wanted the probability of being within 1.0 minute of the mean to be
95%, what is the minimum sample size that would be required? You can
read this off the graph, or solve the appropriate formula for sample size.
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(c) 1998, all rights reserved
Golde I Holtzman
Department of Statistics
Virginia Tech (VPI)
URL: ../STAT5605/sampexer.html
Last updated 10/7/2009
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