Statistics 101: Section L - Laboratory 8
Drawing samples from populations requires some method for random selection. In three group activities
today you will use different methods to randomly select a sample from a population. In the process you will
also learn something about the variability introduced by random sampling.
1.
Group Activity 1:
The first population that we will sample from is the U.S. Senate (1994) edition. There are 100 members.
Some of the characteristics of the Senators that may be of interest are:
name
sex
party
state
years
=
=
=
=
=
Senator’s last name
M for male, F for female
D for Democrat, R for Republican
two letter state abbreviation
years of service
The basic idea of random sampling from a population is to put all of the population members into a
bag, mix them up and draw from the bag. We can’t throw all the Senators into a bag and mix them
up, but we can put laminated slips of paper, one for each Senator, into a bag and use them to represent
the Senators. Use a bag containing 100 laminated slips to perform the sampling described below and
answer the questions.
(a) Choose a sample of 10 U.S. Senators. Record the Senator’s name, sex, party and years of service
on the group answer sheet. Also record the proportion of males, the proportion of Democrats and
the mean years of service for the sample.
(b) Repeat part (a) to obtain two more samples of 10. Be sure to return the 10 sampled slips
to the bag and mix thoroughly before drawing each sample of 10. Again record the
proportion of males, proportion of Democrats and sample mean years of service.
(c) Did the three samples produce the same summary values, i.e. proportions and means? Why?
(d) In the 1994 Senate there are 93 males and 7 females, 54 Democrats and 46 Republicans, and the
mean years of service of 12.4 years. Which sample comes closest to matching the true proportions
in the population?
(e) Since most of the time we take only one sample, what can be said about that a single sample’s
ability to match the population?
2.
Group Activity 2:
Your group will need a sheet of 100 random rectangles for this activity. DO NOT study the sheet
of random rectangles. The random rectangles represent a generic population of values. The areas
of the rectangles could represent the number of people in households or the number of hours per week
students spend on STAT 101. What we wish to investigate in this activity is what can be learned
about the area of the rectangles using sampling techniques.
(a) Each individual in the group should look at the sheet of random rectangles for a few seconds and
guess the average area of the rectangles on the sheet. The unit of measure is the background
square. Thus rectangle 7 has area 4x3=12. Compare your guess with those of the other members
in your group. Discuss why you got different guesses. Record the individual guesses on your group
data sheet.
(b) Each individual should now select five rectangles that, in her/his judgment, are representative of
the rectangles on the sheet. Write down the area for each of the five. Compute the average of the
five areas. How does this compare with your guess? How does it compare with the averages of
the judgment samples for the others in your group? Again discuss why you got different averages.
Record the judgment sample averages on your group data sheet.
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(c) We want to take a random sample from the 100 rectangles. We could cut out the rectangles,
laminate them, put them in a bag, mix thoroughly and draw from the bag. Do you think this
would give you a random sample? Hint: What is true about the slips for the Senators that is not
true of the rectangles?
(d) You now have a 10 sided die available to use. Devise a plan to produce a random sample from
this population using the 10-sided die.
(e) Using your sampling plan, take a simple random sample of size 5 from the rectangles. Record the
rectangle and the area on your group answer sheet. What is the mean area of your sample? How
does it compare with your guess and your judgment sample average?
(f) If you select 5 new random numbers will you come up with the same average area? Explain briefly.
(g) Which do you think will be a better guess of the true average area of the 100 rectangles, the
random sample average of 5 rectangles or the random sample average of 20 rectangles? Explain
briefly.
3.
Group Activity 3:
Pine forests are periodically infested with beetles and the infestations often seem to jump from tree to
tree in a random manner. How can a forester estimate the total number of trees infested with beetles
in a large forest? By taking a sample of plots in the forest and counting the number of infested trees
in the sample, a forester will be able to use the sample data to estimate the total. To simulate this
activity, imagine that Figure 1 represents a forest and the black dots represent infested trees in that
forest. The forest has been divided into 100 plots of equal area.
(a) You have a table of random numbers available for you to use. Devise a plan to produce a random
sample from this population using the random numbers in the table.
(b) Use your sampling plan to randomly select 10 plots from the table. For each plot count the
number of infested trees (dots) that are entirely with in the plot. Do not count trees (dots) that
overlap the edges of a plot. For example, the (Row 0, Column 0) plot has 6 infested trees.
(c) Calculate the mean and the standard deviation of the 10 counts.
(d) Using the mean in (c), estimate the total number of infested trees in the forest.
(e) Again use the random number table to select 10 plots. This time for each plot count the number
of infested trees (dots) that have any part within the plot. For example, the (Row 0, Column 0)
plot has 7 infested trees.
(f) Calculate the mean and standard deviation of the 10 counts.
(g) Using the mean in (f), estimate the total number of infested trees in the forest.
(h) What would be the best estimate of the total number of infested trees in the forest. Briefly explain
your choice.
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Statistics 101: Section L - Laboratory 8
Group Answer Sheet
Names of Group Members:
1.
Group Activity 1:
Sample 1
name
sex
party
years
summaries
p̂M =
p̂D =
X=
Sample 2 summaries
p̂M =
p̂D =
X=
Sample 3 summaries
p̂M =
p̂D =
X=
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2.
Group Activity 2:
Record your group’s guessed values, judgment sample averages and random sample averages in the
spaces below.
(a) Guessed Values
(b) Judgment sample averages
Comparisons:
(c) Difference between Senator’s and rectangles
(d) Sampling plan
(e) Random sample averages
Comparisons:
(f) Same mean for another sample?
(g) Which is better: 5 or 20
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3.
Group Activity 3:
(a) Sampling Plan
(b) Counts for 10 selected plots, interior trees only.
(c) Mean and standard deviation of the 10 counts.
(d) Estimate of the total number of infested trees.
(e) Counts for 10 selected plots, all trees.
(f) Mean and standard deviation of the 10 counts.
(g) Estimate of the total number of infested trees.
(h) Best estimate of the total number of infested trees and explanation.
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