STAT 410 96-2 Assignment 10 Solutions


  1. The systematic samples gave me the following table of sample means:

    Sample Mean Employment Income Mean Government Income
    1176324.750 1241.625
    2160657.500 2855.625
    3147755.600 2631.000
    4150789.250 295.875


    The grand means are just the column means and I got the 2 Ybars to be $158881.7812 and $1756.0312. The variances asked for in parts 1 and 3 are obtained by taking these numbers away from the columns of sample means above, squaring, summing and dividing by 4. I got the variances to be 124172826.8 and 1092836.70 leading to standard errors of $11143.2862 and $1045.3883. The variances for SRS are given by S^2(1-8/32)/32 and you have to compute the two population variances. I get 2084275858 and 8855182.9 which lead to standard errors of $13978.5858 and $911.1385 so that, as expected systematic sampling is better for employment income (where there is a fairly strong trend) and worse for government income (for which the ordering seems more or less random).


  2. There are two ways to do this question. The variance of ybarbar from a cluster sample of n clusters is S^2_M (1-n/128)/(nM^2) and we want to set this equal to S^2(1-36/1024)/36 and solve for n after replacing S^2 and S^2_M by estimates. The first and easiest way is to take your estimate of S^2, that is s^2, from assignment 3 where you drew a SRS and your estimate of S^2_M to be the sample variance of the 6 cluster totals on the midterm. The second, and better way, is to follow the discussion in class and use the raw data provided to fill in the analysis of variance table for the data, then complete the estimated ANOVA for the population. This method may be better because the desired value of n depends only on the ratio of your estimates of S and S_M. The second method uses a ratio estimate where the top and bottom are probably correlated positively and so may well estimate n better even though the srs estimate of S^2 itself is likely to be more accurate than the cluster sample estimate of S^2 based on the ANOVA table; look back at Question 6.4.


The questions.