A random sample of 5222 permanent dwellings on the entire...

A random sample of 5222 permanent dwellings on the entire Navajo Indian Reservation showed that 1619 were traditional Navajo hogans (Navajo Architecture: Forms, History, Distributions by Jett and Spencer, University of Arizona Press). (a) Let $p$ be the proportion of all permanent dwellings on the entire Navajo Rescrvation that are traditional hogans. Find a point estimate for $p$. (b) Find a $99 \%$ confidence interval for $p$. Give a brief interpretation of the confidence interval. (c) Do you think that $n p > 5$ and $n q > 5$ are satisfied for this problem? Explain why this would be an important consideration.

A random sample of 5222 permanent dwellings on the entire Navajo Indian Reservation showed that 1619 were traditional Navajo hogans (Navajo Architecture: Forms, History, Distributions by Jett and Spencer, University of Arizona Press).
(a) Let $p$ be the proportion of all permanent dwellings on the entire Navajo Rescrvation that are traditional hogans. Find a point estimate for $p$.
(b) Find a $99 \%$ confidence interval for $p$. Give a brief interpretation of the confidence interval.
(c) Do you think that $n p > 5$ and $n q > 5$ are satisfied for this problem? Explain why this would be an important consideration.

Key Concepts

Point Estimate for Proportion

A point estimate for the proportion is calculated as the number of successes (or the characteristic of interest) divided by the total sample size. This estimate provides a single best value for the unknown parameter representing the proportion in the overall population.

Confidence Interval for a Proportion

The confidence interval for a proportion is derived by taking the point estimate and adding and subtracting a margin of error, which is based on the estimated standard error from the normal approximation. In this context, a 99% confidence interval is computed to capture the range within which the true population proportion is likely to fall with a high level of certainty.

Interpretation of Confidence Interval

Interpreting a confidence interval involves understanding that if the same sampling method were repeated many times, approximately 99% of the computed intervals would contain the true proportion. This does not mean there is a 99% probability that the true proportion lies in any one specific interval, but it reflects the long-run reliability of the interval estimation method.

Assumptions of Normal Approximation for Proportions

The conditions np > 5 and n(1-p) > 5 are important to ensure that the sampling distribution of the sample proportion is approximately normal. Meeting these criteria allows us to safely use the normal-based method to calculate the confidence interval, ensuring that the approximation of the true sampling distribution is valid.

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