Question 1: A forest has 5 different types of trees. A researcher wants to sample trees for a study, ensuring each type is proportionally represented. If the forest has 200 oak trees, 150 pines, 100 maples, 50 birches, and 30 spruces, and the researcher plans to select a total of 50 trees, how many trees of each type should be sampled to maintain proportional representation?

Answer: Oak: 50*200/530 ≈ 19, Pine: 50*150/530 ≈ 14, Maple: 50*100/530 ≈ 9, Birch: 50*50/530 ≈ 5, Spruce: 50*30/530 ≈ 3

Question 2: Explain why stratified sampling can provide a more accurate estimate of the average number of flowering plants in different regions of a national park compared to simple random sampling.

Answer: Stratified sampling ensures each region (stratum) is proportionally represented, reducing bias and increasing accuracy by capturing variation within each region.

Question 3: A study environmental scientist divides a lake into 4 zones. The zones have different numbers of fish: Zone 1: 300, Zone 2: 200, Zone 3: 150, Zone 4: 350. If the scientist samples 20 fish from the entire lake using stratified sampling, how many should be sampled from each zone?

Answer: Zone 1: 20*300/1000 = 6, Zone 2: 20*200/1000 = 4, Zone 3: 20*150/1000 = 3, Zone 4: 20*350/1000 = 7

Question 4: Construct a stratified sampling plan for a survey on bird species in a habitat with three strata: forest, grassland, and wetlands, having 120, 80, and 50 bird populations respectively. The total sample size is 30. How many should be sampled from each stratum?

Answer: Forest: 30*120/250 ≈ 14, Grassland: 30*80/250 ≈ 10, Wetlands: 30*50/250 ≈ 6

Question 5: In a study of insect populations, the total population consists of 600 beetles, 400 ants, and 200 butterflies. A researcher wants to sample 60 insects proportionally. How many beetles, ants, and butterflies should be sampled?

Answer: Beetles: 60*600/1200 = 30, Ants: 60*400/1200 = 20, Butterflies: 60*200/1200 = 10

Question 6: Why might stratified sampling be particularly useful when studying plant species across different soil types?

Answer: Because different soil types may support different plant species, stratified sampling ensures each soil type is proportionally represented, capturing variation and improving estimate accuracy.

Question 7: A conservation team divides a wetland into three zones based on vegetation density. The zones have 150, 100, and 50 plant species respectively. They plan to sample 20 plants in total. How many should be sampled from each zone?

Answer: Zone 1: 20*150/300 = 10, Zone 2: 20*100/300 ≈ 7, Zone 3: 20*50/300 ≈ 3

Question 8: Identify a common mistake when students calculate stratified samples and suggest how to avoid it.

Answer: A common mistake is not maintaining the proportions of each stratum, leading to biased samples. To avoid this, always multiply the total sample size by the proportion of each stratum.

Question 9: Given a lake with 4 zones having 250, 150, 100, and 200 fish respectively, if a total of 40 fish are sampled proportionally using stratified sampling, how many fish will be sampled from each zone?

Answer: Zone 1: 40*250/700 ≈ 14, Zone 2: 40*150/700 ≈ 9, Zone 3: 40*100/700 ≈ 6, Zone 4: 40*200/700 ≈ 11

Question 10: Explain how stratified sampling can improve the accuracy of estimating deforestation rates across different forest types.

Answer: Stratified sampling ensures each forest type (stratum) is proportionally sampled, capturing variation within each type and reducing sampling bias, thus providing a more accurate estimate.

Question 11: A researcher divides a grassland into three altitudinal zones with known populations of 100, 150, and 250 birds. If they plan to sample 30 birds proportionally, how many birds from each zone should be sampled?

Answer: Zone 1: 30*100/500 = 6, Zone 2: 30*150/500 = 9, Zone 3: 30*250/500 = 15