Question: There are 602 aerobics and fitness centers in Japan (up from 170 five years ago). Adidas, the European maker of sports shoes and apparel, is very interested in this fastgrowing potential market for its products. As part of a marketing survey, Adidas wants to estimate the average income of all members of Japanese fitness centers. (Members of one such club pay $62 to join and another $62 per month. Adidas believes that the average income of all fitness club members in Japan may be higher than that of the general population, for which census data exist.)

Since travel and administrative costs for conducting a simple random sample of all members of fitness clubs throughout Japan would be prohibitive, Adidas decided to conduct a cluster sampling survey. Five clubs were chosen at random out of the entire collection of 602 clubs, and all members of the five clubs were interviewed. The following are the average incomes (in U.S. dollars) for the members of each of the five clubs (the number of members in each club is given in parentheses): $37,237 (560), $41,338 (435), $28,800 (890), $35,498 (711), $47,446 (230). Give the cluster sampling estimate of the population mean income for all fitness club members in Japan. Also give a 90% confidence interval for the population mean. Are there any limitations to the methodology in this case?