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Chapter 6: Q63E (page 363)
The following is a 90% confidence interval for p:(0.26, 0.54). How large was the sample used to construct thisinterval?
The required sample size is 34.
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Aluminium cans contaminated by fire. A gigantic warehouselocated in Tampa, Florida, stores approximately 60million empty aluminium beer and soda cans. Recently, a fireoccurred at the warehouse. The smoke from the fire contaminatedmany of the cans with blackspot, rendering them unusable.
A University of South Florida statistician was hiredby the insurance company to estimate p,the true proportionof cans in the warehouse that were contaminated by the fire. How many aluminium cans should be randomly sampled toestimate pto within .02 with 90% confidence?
Suppose you want to estimate a population mean,,and,,,and.Find an approximate 95% confidence interval for.
Hospital length of stay. Health insurers and the federal government are both putting pressure on hospitals to shorten the average length of stay (LOS) of their patients. The average LOS in the United States is 4.5 days (Healthcare Cost and Utilization Project Statistical Brief, October 2014). A random sample of 20 hospitals in one state had a mean LOS of 3.8 days and a standard deviation of 1.2 days.
a. Use a 90% confidence interval to estimate the population's mean LOS for the state’s hospitals.
b. Interpret the interval in terms of this application.
c. What is meant by the phrase “90% confidence interval”?
A sampling dispute goes to court. Sampling of Medicare and Medicaid claims by the federal and state agencies who administer those programs has become common practice to determine whether providers of those services are submitting valid claims. (See the Statistics in Action for this chapter.) The reliability of inferences based on those samples depends on the methodology used to collect the sample of claims. Consider estimating the true proportion, p, of the population of claims that are invalid. (Invalid claims should not have been reimbursed by the agency.) Of course, to estimate a binomial parameter, p, within a given level of precision we use the formula provided in Section 6.5 to determine the necessary sample size. In a recent actual case, the statistician determined a sample size large enough to ensure that the bound on the error of the estimate would not exceed 0.05, using a 95% confidence interval. He did so by assuming that the true error rate was, which, as discussed in Section 6.5, provides the maximum sample size needed to achieve the desired bound on the error.
a. Determine the sample size necessary to estimate p to within .05 of the true value using a 95% confidence interval.
b. After the sample was selected and the sampled claims were audited, it was determined that the estimated error rate was and a 95% confidence interval for p was (0.15, 0.25). Was the desired bound on the error of the estimate met?
c. An economist hired by the Medicare provider noted that, since the desired bound on the error of .05 is equal to 25% of the estimated invalid claim rate, the “true” bound on the error was .25, not .05. He argued that a significantly larger sample would be necessary to meet the “relative error” (the bound on the error divided by the error rate) goal of .05, and that the statistician’s use of the “absolute error” of .05 was inappropriate, and more sampling was required. The statistician argued that the relative error was a moving target, since it depends on the sample estimate of the invalid claim rate, which cannot be known prior to selecting the sample. He noted that if the estimated invalid claim rate turned out to be larger than .5, the relative error would then be lower than the absolute error bound. As a consequence, the case went to trial over the relative vs. absolute error dispute. Give your opinion on the matter. [Note: The Court concluded that “absolute error was the fair and accurate measure of the margin of error.” As a result, a specified absolute bound on the error continues to be the accepted method for determining the sample size necessary to provide a reliable estimate of Medicare and Medicaid providers’ claim submission error rates.]
General health survey. The Centres for Disease Control and Prevention (CDCP) in Atlanta, Georgia, conducts an annual survey of the general health of the U.S. population as part of its Behavioural Risk Factor Surveillance System. Using random-digit-dialing, the CDCP telephones U.S. citizens over 18 years of age and asks them the following four questions:
1. Is your health generally excellent, very good, good, fair, or poor?
2. How many days during the previous 30 days was your physical health not good because of injury or illness?
3. How many days during the previous 30 days was your mental health not good because of stress, depression, or emotional problems?
4. How many days during the previous 30 days did your physical or mental health prevent you from performing your usual activities?
Identify the parameter of interest for each question
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