It is desired to estimate the average consumer spending on durable goods, annually, in a defined geographical region. Spending patterns are influenced by disposable income. Based on county tax records, the population has been segmented into three groups: (1) annual income < $50,000; (2) between $50,000 and $100,000; (3) over $ 100,000. The number of people in each category has been identified as 2000,7000, and 1000, respectively. Sampling costs limit the total sample size to be 200.
(a) Using a stratified proportional sample, what should the sample sizes be from each group?
(b) Using the sample sizes found in part (a), the following sample means and standard deviations (in thousands of dollars) were found: 1 = 3.5, s1 = 1.2; 2 = 7.6, s2 = 2.8; 3 = 15.1, s3 = 6.4. Estimate the mean and standard deviation of the amount spent on durables.
(c) Find a bound on the error of estimation using a 95% level of confidence.
(d) Suppose that you had selected a simple random of the same total size from the population and that the results indicated the following sample mean and standard deviation: = 6.8, s = 5.6. Find a bound on the error of estimation using a 95% level of confidence.
(e) What is the percentage improvement in precision by using a stratified proportional sample relative to a simple random sample?