Two dentists conduct a survey on the condition of teeth of 200 children in a village. The first dentist selects using simple random sampling 20 children among the 200, and counts the data in the sample according to the number of teeth with cavities. The results are presented in the following table. The second dentist examines the 200 children but with the solo goal of determining who has no cavities. He notices that 50 children are in the category.
Number of teeth with cavities 0 1 2 3 4 5 6 7 8 Number of children 8 4 2 2 1 2 0 0 1
a. Estimate the mean number of teeth with cavities per child in the village using only results of the first dentist. What is the variance of the unbiased estimator? Estimate this variance and the associated 95% confidence interval.
b. Propose another estimator for the mean number of teeth with cavities per child using the results of the two dentists. Calculate the new estimate, and estimate the gain in the efficiency obtained.