What is the probability of selecting the sample

Assignment Help Basic Statistics

Reference no: EM131394497

Q1. Consider a hypothetical population of size 8 with an auxiliary variable x that has known values given by x₁ = 3, x₂ = 7, x₃ =16, x₄ = 20, x₅ = 26, x₆ = 33, x₇ = 40, x₈ = 55. To select a sample of size 3 from this population, a unit is first drawn with probability proportional to the value of x₁, and then two more units are drawn from the remaining 7 units using simple random sampling.

(a) What is the probability of selecting the sample {2, 5, 7} under this sampling scheme? Hint: There are 3 possibilities, depending on which unit is drawn first. For subsequent draws, you can use the facts that the first and second order inclusion probabilities under simple random sampling are given by n/N and n(n -1)/N(N-1), where n is the sample size, and N the population size.

(b) What is the probability that unit 7 is included in the sample?

Q2. A first phase simple random sample S of size n (with sample mean x^- for variable x) is drawn from a finite population U of size N with population mean x^-_U and population variance S²_x for variable x. At phase 2, a subsample S₂ of size n₂ < n is drawn from the phase 1 samples using S simple random sampling. It follows naturally that S₂ is a simple random sample of size n₂ from population U (no need to prove this).

(a) Let x^-₂ be the mean of x calculated from subsample S₂, write down E(x^-₂) and var(x^-₂).

(b) Show that E((x^-₂ - x^-)x^-)= 0 . You may use the fact that E((x^-₂- x^-)x^-)= E{E₂((x^-₂-x^-)x^-)} where E₂ denote the conditional expectation under the second phase sampling given the phase 1 sample.

(d) Use (c) to show that var(x^-₂ - x^-) = var(x^-₂) - var(x^-) and simplify it to (1/n² - 1/n)S²_x.

Reference no: EM131394497

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