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A forest manager uses a 10 × 10 grid of a forest plot to estimate the number of white pine trees and red pine trees on the given plot of land. There are 100 sections in the grid, shown on the next two slides. White pine trees are noted with “w” and red pine trees are noted with “r.”
The forest manager randomly selects the following 10 sections out of the 100 sections of the forest (shown in bold on the grid):
B–10 E–3 F–1 C–3 I–9
A–3 H–10 D–9 F–9 B–6
Use this information to answer the questions that follow.
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1.3.3: Other Methods of Random Sampling
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1.3.3: Other Methods of Random Sampling
A B C D E F G H I J
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1.3.3: Other Methods of Random Sampling
A B C D E F G H I J
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1. Determine the number of white pines and red pines in the sample sections.
2. Estimate the number of white pines and red pines in the forest plot.
3. Explain how a simple random sample could be used in this situation.
4. Why is a simple random sample not practical in this situation?
5. Does the sampling procedure provide a representative sample? Explain your answer.
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1.3.3: Other Methods of Random Sampling
1. Determine the number of white pines and red pines in
the sample sections.
• Use the grid to determine the number of white pines
and red pines in each sample selection.
• Count the occurrences of white pine trees, w, and
red pine trees, r, for each section.
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1.3.3: Other Methods of Random Sampling
• Section B–10 has 2 white pine trees and 6 red pine trees.
• Section E–3 has 3 white pine trees and 5 red pine trees.
• Section F–1 has 4 white pine trees and 4 red pine trees.
• Section C–3 has 5 white pine trees and 3 red pine trees.
• Section I–9 has 5 white pine trees and 3 red pine trees.
• Section A–3 has 1 white pine tree and 6 red pine trees.
• Section H–10 has 3 white pine trees and 5 red pine trees.
• Section D–9 has 3 white pine trees and 5 red pine trees.
• Section F–9 has 5 white pine trees and 3 red pine trees.
• Section B–6 has 3 white pine trees and 4 red pine trees. 7
1.3.3: Other Methods of Random Sampling
• To find the number of white pine trees and red pine trees in the sample, find the total number of white pine trees and the total number of red pine trees.
white pine trees = 2 + 3 + 4 + 5 + 5 + 1 + 3 + 3 + 5 + 3 = 34
• The total number of white pine trees is 34.
red pine trees = 6 + 5 + 4 + 3 + 3 + 6 + 5 + 5 + 3 + 4 = 44
• The total number of red pine trees is 44.
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1.3.3: Other Methods of Random Sampling
2. Estimate the number of white pines and red pines in the forest plot.
• Since trees were counted in 10 of the 100 sections, we can use proportions to estimate the number of trees in the entire plot.
• Set up a proportion to estimate the total number of white pines for the entire plot.
• Use the number of sections sampled (10) for the sample size, and the number of sections in the entire forest plot (100) for the population size.
• Create a proportion for white pines, as shown on the next slide; then, solve it for the unknown variable.
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1.3.3: Other Methods of Random Sampling
Substitute known values.
10w = 3400 Cross-multiply to solve
the proportion.
w = 340 Divide both sides by 10.
• Based on the sample, there are estimated to be 340 white pines in the forest plot. 10
1.3.3: Other Methods of Random Sampling
• Repeat the process to estimate the number of red pines.
• Create a proportion for red pines, as shown on the next slide; then, solve it for the unknown variable.
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1.3.3: Other Methods of Random Sampling
Substitute known values.
10r = 4400 Cross-multiply to solve the proportion.
r = 440 Divide both sides by 10.
• Based on the sample, there are estimated to be 440 red pines in the forest plot.
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1.3.3: Other Methods of Random Sampling
3. Explain how a simple random sample could be used in this situation. • To use a simple random sample in this situation,
every tree would need to be labeled with a number, and then random integers could be generated to correspond to the trees that are selected in the sample.
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1.3.3: Other Methods of Random Sampling
4. Why is a simple random sample not practical in this situation? • It would be extremely time-consuming to label
every tree. Also, if the goal is to estimate the number of trees, then the time it takes to label them is more than the time it would take to count them.
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1.3.3: Other Methods of Random Sampling
5. Does the sampling procedure provide a representative sample? Explain your answer. • Since each of the 100 sections has an equal
chance of selection, the sample is representative. It is not a simple random sample and the estimate of the number of trees will contain sampling error, but the sample is not biased.
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1.3.3: Other Methods of Random Sampling
