Diet Habits A federal report indicated that 27 % of children...

Diet Habits A federal report indicated that $27 \%$ of
children ages 2 to 5 years had a good diet-an increase
over previous years. How large a sample is needed to
estimate the true proportion of children with good diets
within $2 \%$ with $95 \%$ confidence?

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Key Concepts

Sample Size Calculation

Sample size calculation for estimating proportions involves using a formula that relates the desired margin of error, the confidence level (typically via the z-score), and the estimated proportion. This calculation ensures that the sample is large enough to achieve the specified level of precision in the estimate.

Normal Approximation

The normal approximation, grounded in the Central Limit Theorem, allows the use of z-scores to approximate the distribution of sample proportions when the sample size is sufficiently large. This is critical for constructing confidence intervals and calculating sample sizes for proportion estimates.

Margin of Error

The margin of error quantifies the maximum expected difference between the sample estimate and the true population parameter. It is influenced by the variability in the data and the chosen confidence level, and it determines how precise the estimate will be.

Confidence Interval

A confidence interval provides a range of values that is likely to contain the true population parameter. It reflects the degree of uncertainty or certainty in a sampling method and is associated with a specific confidence level, such as 95%.

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