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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 group. $$ \begin{array}{ccc} \hline & \text { control } & \text { treatment } \\ \hline \text { alive } & 4 & 24 \\ \text { dead } & 30 & 45 \\ \hline \end{array} $$ Suppose we are interested in estimating the difference in survival rate between the control and treatment groups using 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

Understanding the Normal Approximation

The normal approximation is used to estimate the differences between two proportions when both samples are large enough. Generally, if both the number of successes and failures (alive and dead in this case) in each group are above 5, we can use the normal approximation to the binomial distribution.

02

Checking the Conditions for the Control Group

In the control group, there are 4 survivors (alive) and 30 non-survivors (dead). Since 4 < 5, the number of successes is less than 5, which does not satisfy the normal approximation condition.

03

Checking the Conditions for the Treatment Group

In the treatment group, 24 patients survived and 45 did not. Although both the number of alive and dead instances exceeds 5 here, for the confidence interval to be valid under normal approximation, it must also hold true for the control group, which it does not.

04

Evaluating the Implications of Using the Normal Approximation

If we proceeded with constructing the confidence interval using the normal approximation despite not meeting the necessary conditions, it could lead to inaccurate estimation of confidence bounds. This is because the standard error may be underestimated, increasing the chance of a Type I error (incorrectly rejecting a true null hypothesis).

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Key Concepts

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

Confidence Interval

A confidence interval is a range of values, derived from sample data, that is likely to contain the true value of an unknown population parameter. Confidence intervals provide a way to estimate that population parameter, reflecting the degree of uncertainty around the sample estimate.

When calculating a confidence interval, you determine a range that you are confident contains the true parameter, often expressed as a percentage like 95% or 99%. This percentage is called the confidence level, and a 95% confidence level means you can be 95% certain the interval contains the true parameter.

• For example, if you are estimating the difference in survival rates between two groups (such as those in a heart transplant study), you might calculate a confidence interval around this difference in survival rates.
• The width of the confidence interval provides information about the precision of the estimate—narrower intervals indicate more precise estimates.

One key point to remember while constructing confidence intervals is to ensure that the underlying data satisfies necessary conditions for the statistical method chosen, otherwise, the interval may not accurately reflect the uncertainty around the estimate.

Normal Approximation

Normal approximation is a method used to simplify calculations when dealing with binomial distributions, especially when sample sizes are large. The idea is to approximate the binomial distribution with a normal distribution because the normal distributions are simpler to work with mathematically.

The normal approximation becomes valid under specific conditions. The rule of thumb is that both the number of "successes" and the number of "failures" (in this case, alive and dead patients) should each be at least 5. This ensures that the binomial distribution is sufficiently close to a normal distribution.

• When these conditions are met, the normal distribution can then be used as an approximation to calculate probabilities and confidence intervals more easily.
• If these conditions are not met, the approximation could yield misleading results.

Using the normal approximation where conditions are not met may lead to inaccurate confidence intervals and increased risk of statistical errors, such as Type I or Type II errors.

Binomial Distribution

The binomial distribution is a discrete probability distribution that models the number of successes in a fixed number of trials, where each trial has only two possible outcomes—success or failure. In the context of the heart transplant study, "alive" and "dead" can be considered as the two outcomes.

The distribution is defined by two parameters: the number of trials (n) and the probability of success in each trial (p). It answers questions such as "How many patients survive out of the total?" or "How often does a treatment result in survival?"

• Calculating probabilities using the binomial distribution involves combinations (ways of arranging successes out of total trials) and individual success probabilities.
• This type of distribution is particularly useful because it describes many natural phenomena in binary outcomes, providing a foundation for more complex statistical constructs, such as normal approximation.

Despite its usefulness, utilizing the binomial distribution directly can become unwieldy for large sample sizes, which is why approximations, like the normal approximation, are often employed—provided certain conditions are met.