To estimate the number of typographical errors in a 65-page manuscript, a systematic sample of pages is selected by first selecting a random number between 1 and 10 and including in the sample that numbered page and every 10th page thereafter. The random number selected was 6. The number of typographical errors on the sample pages were 1, 0, 2, 3, 0, and 1. Assume that no errors on sample pages were missed.
(a) Give an unbiased estimate, under the design used, of the total number of errors in the manuscript. What design was used?
(b) The person doing the survey estimated the total number of errors in the manuscript by 65(1 + 0 + 2 + 3 + 0 + 1)/6 = 75.83. Which estimator was used? Is it unbiased with the design used?
(c) The variance of the estimator was estimated by 65(65 - 6)(1.37)/6, where 1.37 is the sample variance of the six error counts. Is this unbiased for the actual variance of the estimator of the total number of errors? Discuss.