Stratified Sampling

Given a population divided into three strata with sizes 100, 300, and 600, and standard deviations 5, 10, and 15, respectively, how many individuals should be sampled from the first stratum if a total sample size of 120 is desired using Neyman’s optimal allocation?

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Answer explanation

To find the optimal allocation, calculate the square root of each stratum size multiplied by its standard deviation. Then, divide each value by the sum of all values. For the first stratum, this results in 100 / (100 + 300 + 600) * 120 = 10 individuals.

A population is divided into three strata with sizes 200, 300, and 500. If a sample of 100 is to be drawn using proportional allocation, how many individuals should be sampled from the second stratum?

• A. 20

• B. 25

• C. 30

• D. 35

Answer explanation

Proportional allocation involves sampling in proportion to stratum size. For the second stratum, 300/1000 * 100 = 30 individuals should be sampled. Therefore, the correct choice is C. 30.

A population is divided into two strata of sizes 200 and 800 with standard deviations 6 and 12, respectively. If the total sample size is 100 using optimal allocation, how many samples should be drawn from the second stratum?

• A. 70

• B. 75

• C. 80

• D. 85

Answer explanation

To find the optimal allocation, use the formula: n1 = N1 * (s1/s)^2 and n2 = N2 * (s2/s)^2. Substituting the values, n2 = 800 * (12/18)^2 = 80. Therefore, 80 samples should be drawn from the second stratum.

In a stratified random sample, the sample sizes are 50, 100, and 150 from three strata with proportions 0.1, 0.2, and 0.3, respectively. What is the estimated population proportion?

• A. 0.15

• B. 0.20

• C. 0.25

• D. 0.30

Answer explanation

The estimated population proportion is calculated by taking the weighted average of the strata proportions based on their sample sizes. (0.1*50 + 0.2*100 + 0.3*150) / (50 + 100 + 150) = 0.25, so the correct answer is C. 0.25.

A researcher divides a population into four strata with sizes 150, 250, 300, and 300. If the researcher decides to take a sample of 200 using proportional allocation, how many individuals should be sampled from the first stratum?

• A. 20

• B. 25

• C. 30

• D. 35

Answer explanation

Using proportional allocation, the number of individuals sampled from each stratum is proportional to its size. For the first stratum with a size of 150, the number sampled would be (150/1000) * 200 = 30.

If the population proportion of a certain characteristic is 30%, and a post-stratified sample has 60 out of 200 individuals exhibiting that characteristic, what is the post-stratified estimate of the total population exhibiting that characteristic?

• A. 45

• B. 50

• C. 60

• D. 70

Answer explanation

The post-stratified estimate is calculated by multiplying the population proportion (30%) by the total sample size (200), resulting in 60 individuals exhibiting the characteristic.

A researcher uses optimal allocation for a population divided into two strata of sizes 250 and 750. The standard deviations are 10 and 20, respectively. If the total sample size is 200, what is the sample size for the first stratum?

• A. 25

• B. 40

• C. 50

• D. 60

Answer explanation

The sample size for the first stratum can be calculated using the formula: n1 = N1/N * n, where N1 is the size of the first stratum, N is the total population size, and n is the total sample size. Substituting the given values, n1 = (250/1000) * 200 = 50.

A researcher stratifies a population into two strata. The first stratum has a size of 400 with a mean income of $30,000 and the second stratum has a size of 600 with a mean income of $40,000. What is the combined mean income?

• A. $35,000

• B. $36,000

• C. $37,000

• D. $38,000

Answer explanation

To find the combined mean income, calculate the weighted average: (400*$30,000 + 600*$40,000) / (400 + 600) = $36,000. Therefore, the correct choice is B. $36,000.

A population is divided into three strata with sizes 200, 300, and 500. The proportions of a characteristic are 0.2, 0.3, and 0.4, respectively. What is the combined proportion estimate for the population?

• A. 0.30

• B. 0.32

• C. 0.34

• D. 0.36

Answer explanation

To find the combined proportion estimate, multiply the strata sizes by their respective proportions, then sum the results. (200*0.2 + 300*0.3 + 500*0.4) / (200 + 300 + 500) = 0.32. Therefore, the correct choice is B. 0.32.

A population is divided into four strata with sizes 100, 200, 300, and 400. If a sample of 150 is to be drawn using proportional allocation, how many individuals should be sampled from the third stratum?

• A. 30

• B. 40

• C. 50

• D. 60

Answer explanation

To find the number of individuals to sample from the third stratum, calculate 150 * (300/1000) = 45. Therefore, the correct choice is B. 40.