Get High Airlines wants to estimate the fraction of smokers...

Get High Airlines wants to estimate the fraction of smokers $p$ among their passenger population. The airline plans to use sampling with replacement to determine their estimate $\hat{p}$ of $p$. They set up the requirement that $$ P[|\hat{p}-p| \geq 0.005] \leq 0.05 \text {. } $$ (a) How large a sample should Get High take if nothing is assumed about the size of $p$ ? (b) How large a sample is required if Get High knows that $p$ is very close to 0.4 ?

Get High Airlines wants to estimate the fraction of smokers $p$ among their passenger population. The airline plans to use sampling with replacement to determine their estimate $\hat{p}$ of $p$. They set up the requirement that
$$
P[|\hat{p}-p| \geq 0.005] \leq 0.05 \text {. }
$$
(a) How large a sample should Get High take if nothing is assumed about the size of $p$ ?
(b) How large a sample is required if Get High knows that $p$ is very close to 0.4 ?

Key Concepts

Normal Approximation for the Binomial Distribution

When the sample size is large, the sampling distribution of a proportion can be approximated by a normal distribution. This approximation facilitates calculating the required sample size by relating the margin of error, the standard error of the sample proportion, and the z-score corresponding to the desired confidence level. It is especially useful in problems like this where probability constraints on the estimation error are specified.

Proportion Estimation

This concept involves using sample data to estimate the proportion of a population that has a certain attribute. In the context of the problem, the aim is to estimate the fraction of smokers among airline passengers. The estimator used is the sample proportion, which is subject to sampling variability and whose precision depends on the sample size and the true proportion.

Margin of Error

The margin of error represents the maximum allowable difference between the sample estimate and the true proportion. In the problem, the airline sets a tolerance (0.005) for this difference with a specified probability. The margin of error is crucial in relating the sample size to the desired precision of the estimate.

Confidence Level and Error Probability

This concept pertains to the probability that the true proportion lies within the specified margin of error of the estimate. The problem requires that the probability of an error exceeding the margin of error is at most 0.05, which is analogous to a confidence level in interval estimation. It connects the sample size requirement to a statistical guarantee about the estimator’s performance.

Maximum Variance Principle

For a Bernoulli random variable, which models a proportion, the variance is given by p(1-p) and is maximized when p = 0.5. This concept is employed in sample size determination when no prior estimate of p is available, ensuring a conservative (largest) sample size necessary to meet the error probability requirement.

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