Case studies showed that out of 10,351 convicts who escaped...

Case studies showed that out of 10,351 convicts who escaped from U.S. prisons, only 7867 were recaptured (The Book of Odds by Shook and Shook, Signet). (a) Let $p$ represent the proportion of all escaped convicts who will eventually be recaptured. Find a point estimate for $p$. (b) Find a $99 \%$ confidence interval for $p$. Give a brief statement of the meaning of the confidence interval. (c) Is use of the normal approximation to the binomial justified in this problem? Explain.

Case studies showed that out of
10,351 convicts who escaped from U.S. prisons, only 7867 were recaptured
(The Book of Odds by Shook and Shook, Signet).
(a) Let $p$ represent the proportion of all escaped convicts who will eventually be recaptured. Find a point estimate for $p$.
(b) Find a $99 \%$ confidence interval for $p$. Give a brief statement of the meaning of the confidence interval.
(c) Is use of the normal approximation to the binomial justified in this problem? Explain.

Key Concepts

Interpreting Confidence Intervals

Interpreting a confidence interval involves understanding that it estimates the range in which the true parameter lies with a certain level of confidence, such as 99%. This means that if the same sampling process was repeated many times, approximately 99% of these intervals would contain the true population parameter.

Normal Approximation to the Binomial Distribution

When dealing with binomial proportions, the normal approximation is used to approximate the binomial distribution with a normal distribution, provided that the sample size is large enough such that both np and n(1-p) are sufficiently large. This approximation simplifies the process of computing confidence intervals and hypothesis tests.

Point Estimation

Point estimation involves deriving a single best estimate for an unknown parameter based on sample data. In the context of proportions, the sample proportion (obtained by dividing the number of successful cases by the total number of trials) serves as the point estimate for the true population proportion.

Confidence Interval for Proportions

A confidence interval for a proportion provides a range of values that is likely to contain the true population proportion with a specified level of confidence. It is constructed using the point estimate and its standard error, and it communicates the uncertainty inherent in the estimation process.

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