91Ó°ÊÓ

The article Hospitals Dispute Medtronic Data on Wires" (The Wall Street journal. February 4, 2010) describes several studies of the failure rate of defibrillators used in the trearment of heart problems. In one study conducted by the Mayo Clinic, it was reported that failures were experienced within the first 2 years by 18 of 89 patients under 50 years old and 13 of 362 paticnts age 50 and older who received a particular type of defibrillator. Assume it is reasonable to regard these two samples as representative of patients in the two age groups who receive this type of defibrillator. a. Construct and interpret a \(95 \%\) confidence interval for the proportion of patients under 50 years old who experience a failure within the first 2 years after receiving this type of defibrillator. b. Construct and interpret a \(99 \%\) confidence interval for the proportion of patients age 50 and older who expericnce a failure within the first 2 years after receiving this type of defibrillator. c. Suppose that the researchers wanted to estimate the proportion of patients under 50 years old who experience a failure within the first 2 years after receiving this type of defibrillator to wirhin .03 with \(95 \%\) confidence. How large a sample should be used? Use the results of the study as a preliminary estimate of the population proportion.

Step by step solution

01

Calculate 95% Confidence Interval for patients under 50

The proportion of patients under 50 who experience a failure is calculated as \(\frac{18}{89}\). A \(95\%\) confidence interval is given by \(\hat{p} ± 1.96 * \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}\), where \(\hat{p}\) is the estimated proportion and n is the sample size. Substituting the given values yields the confidence interval.

02

Calculate 99% Confidence Interval for patients 50 and older

The proportion of patients 50 and older who experience a failure is calculated as \(\frac{13}{362}\). A \(99\%\) confidence interval is given by \(\hat{p} ± 2.576 * \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}\), where \(\hat{p}\) is the estimated proportion and n is the sample size. Substituting the given values yields the confidence interval.

03

Determine Sample size for patients under 50

The formula for calculating the sample size needed to estimate the proportion within a certain interval with a given level of confidence is \(n = \frac{Z^2_α/2*P(1-P)}{E^2}\), where \(Z_α/2\) is the Z value for the desired confidence level, P is the estimated proportion and E is the margin of error. By substituting the values from the study and the given margin of error of 0.03 into this formula, we can calculate the necessary sample size.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Statistics

At its core, statistics is the branch of mathematics that deals with collecting, analyzing, interpreting, presenting, and organizing data. It allows us to make sense of vast amounts of information and to make informed decisions based on that information. In our exercise, we dive into one of the many statistical tools: the confidence interval. This is used to estimate the reliability of an estimate. For instance, a confidence interval for the proportion of patients experiencing a device failure gives us a range in which the true proportion is likely to lie with a certain level of confidence, typically expressed as 90%, 95%, or 99%.

When interpreting a confidence interval, it's important to understand that the wider the interval, the more uncertainty there is about the proportion estimate. However, a wider interval also means that there's a higher probability that the interval contains the true proportion. For example, the range given by our 95% confidence interval for patients under 50 means that we can be 95% sure that the true failure rate falls within that range.

Proportion Estimation

Proportion estimation involves determining the fraction of a population that has a specific attribute. In the case of the defibrillator study, we're interested in the proportion of patients who experienced a device failure within a certain time frame. To estimate this proportion, we use the sample proportion as an estimate for the population proportion. This sample proportion, often denoted as \(\hat{p}\), is calculated simply by dividing the number of observed successes (in this case, device failures) by the total number of observations.

Once the sample proportion is calculated, we can create a confidence interval around it, which gives us an estimated range of where we believe the true population proportion lies. The key is to have a representative sample because if the sample does not accurately reflect the population, the confidence interval may not be valid. This is why in the exercise, it's assumed that the sampled patients are representative of the entire population receiving the defibrillator.

Sample Size Determination

Determining the correct sample size is critical for ensuring that statistical estimates are as accurate as possible. The size of the sample affects the precision of our estimates—too small a sample could lead to wide confidence intervals, while needlessly large samples may waste resources. The exercise's part (c) addresses this concern. It asks how large a sample is needed to estimate the proportion of patients under 50 years old within a 3% margin of error with 95% confidence.

To find this, statisticians use a formula that incorporates the level of confidence (reflected in the Z value), the estimated proportion based on preliminary data (which can be improved with more accurate pilot studies), and the desired margin of error. In practice, this means that if we want to be highly confident in our estimate and have a small margin of error, we will need a larger sample size. Thus, accurate planning for sample size is a fundamental step before conducting a study.