There are k interviewers and they are each different in their manner of interviewing and hence may obtain slightly different responses. To make the notation simple, we assume that each interviewer conducts the same number of interviews. Let n denote the total sample size and n = k * m. There are k subsamples and each interviewer will be assigned m subjects.

Objective: to use simple random sampling to estimate \(\mu\)

• Interviewer \(1-y_{11}, y_{12}, y_{13},...,y_{1m}\)
• Interviewer \(2-y_{21}, y_{22}, y_{23},...,y_{2m}\)
• Interviewer \(3-y_{31}, y_{32}, y_{33},...,y_{3m}\)

• Interviewer \(k-y_{k1}, y_{k2}, y_{k3},...,y_{km}\)

The average for the ith interviewer is denoted as:

\(\bar{y}_i=\dfrac{1}{m}\sum\limits_{j=1}^m y_{ij}\)

The grand average is denoted as:

\(\bar{y}=\dfrac{1}{k}\sum\limits_{i=1}^k \bar{y}_i\)

The grand average \(\bar{y}\) is unbiased for μ and the estimated variance of \(\bar{y}\) is:

\(\hat{V}ar(\bar{y})=\dfrac{N-n}{N}\cdot \dfrac{s^2_k}{k}\)

\(\text{where } s^2_k=\dfrac{\sum\limits_{i=1}^k (\bar{y}_i-\bar{y})^2}{k-1}\)

The technique of interpenetrating the subsample gives an estimate of the variance of ybar that accounts for interviewer biases. In practice, the estimated variance given in the above formula is usually larger than the estimate of the variance by using simple random sampling.