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Heart transplant success. The Stanford University Heart Transplant Study was conducted to determine whether an experimental heart transplant program increased lifespan. Each patient entering the program was officially designated a heart transplant candidate, meaning that he was gravely ill and might benefit from a new heart. Patients were randomly assigned into treatment and control groups. Patients in the treatment group received a transplant, and those in the control group did not. The table below displays how many patients survived and died in each 41 group. \begin{tabular}{ccc} \hline & control & treatment \\ \hline alive & 4 & 24 \\ dead & 30 & 45 \\ \hline \end{tabular} A hypothesis test would reject the conclusion that the survival rate is the same in each group, and so we might like to calculate a confidence interval. Explain why we cannot construct such an interval using the normal approximation. What might go wrong if we constructed the confidence interval despite this problem?

Step by step solution

01

Understand Normal Approximation Applicability

The normal approximation is suitable when the sample size is large enough, typically with both success and failure counts in each group being at least 5. This ensures that the binomial distribution can be approximated by the normal distribution, which is crucial for calculating standard normal confidence intervals.

02

Check the Sample Data for the Control Group

The control group has 4 surviving patients and 30 deceased patients. The small number of survivors (successes) in the control group (4) indicates a potential problem as it doesn't meet the general rule of having at least 5 successes and failures.

03

Analyze the Sample Data for the Treatment Group

For the treatment group, there are 24 survivors and 45 deceased individuals. Both groups seem to have a substantial number of cases for applying the normal approximation. However, it is essential to check the sample size against the approximation rule for caution.

04

Explain Why Normal Approximation Fails

The normal approximation might fail due to the small number of successes (4) in the control group, as this does not satisfy the normal approximation rule of having at least 5 successes and failures, which is essential to ensure a valid approximation.

05

Determine Consequences

Constructing a confidence interval under this condition using normal approximation might lead to inaccurate intervals. This could potentially mislead the inference of whether the difference in survival rates is significant as small sample sizes lead to unreliable approximations.

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Key Concepts

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

Heart Transplant Study

The Stanford University Heart Transplant Study aimed to investigate the effect of heart transplants on extending the lives of patients who were critically ill. This study was meticulously designed to analyze whether those who underwent the procedure had better survival rates compared to those who did not. Participants in the study, all gravely ill, were designated as either "treatment" or "control" groups. In the treatment group, patients received heart transplants. Meanwhile, the control group, unfortunately, did not receive the transplants. By structuring the study this way, researchers hoped to get clear evidence on the efficacy of heart transplants in prolonging life for those with severe heart conditions.
The striking element of this study was its rigor in methodology, using clear outcomes like survival rates to measure success. However, comparing these rates required careful statistical analysis to determine if the difference observed was significant and not due to chance.

Random Assignment

Random assignment is a cornerstone of experimental research, ensuring the reliability and validity of a study’s results. In the Heart Transplant Study, patients were randomly allocated to either the treatment or control group. This randomization process helps eliminate selection bias, ensuring that any differences observed between groups can be attributed more confidently to the treatment itself rather than pre-existing differences between groups.
Random assignment creates equivalence among groups in terms of potential confounding variables. Thus, researchers can be more certain that the observed effects were due to the heart transplant and not some other factor. By balancing known and unknown factors equally across treatment conditions, random assignment strengthens the causal inference of the study.

Normal Approximation

Normal approximation is a statistical technique used to simplify the analysis of binomial distributions, especially when dealing with large sample sizes. It allows researchers to use the normal distribution, which is generally easier to handle mathematically, to estimate the properties of a binomial distribution. This method is particularly useful when both the number of successes and failures in a study meet a minimum threshold, often cited as at least 5.
In the case of the Heart Transplant Study, the issue arises with the control group. There were only 4 survivors, which is less than the 5 that is considered necessary for a reliable normal approximation. When this prerequisite isn't met, the normal approximation could lead to inaccurate results. The approximation may fail to accurately reflect the actual distribution of data, thus potentially leading to misleading conclusions about the effectiveness of heart transplants in this scenario.

Confidence Interval

Confidence intervals provide a range of values that are likely to contain the true effect size or parameter value, in this case, the true difference in survival rates between the treatment and control groups. Constructing a confidence interval involves using the data collected to estimate the particular range within which the true parameter is expected to lie.
However, the construction of confidence intervals relies on the assumptions underlying the chosen statistical methods. For the Heart Transplant Study, using normal approximation under conditions where there are insufficient successes in one group could result in an unreliable confidence interval. An inaccurate interval might suggest certainty where there is less, potentially leading decision-makers to wrongly estimate the effectiveness of heart transplants. Confidence intervals are only meaningful when the underlying statistical assumptions are met; otherwise, they can mislead interpretations of the study’s findings.