Question

a federal report indicated that 23% of children under age 6 live in poverty in arizona, an increase over previous years. how large a sample is needed to estimate the true proportion of children under age 6 living in poverty in arizona within 1% with 99% confidence? round the intermediate calculations to three decimal places and round up your final answer to the next whole number.

n =

Explanation:

Step1: Determine the values of \(p\), \(q\), \(E\), and \(z_{\alpha/2}\)

• Given \(p = 0.23\) (proportion of children under age 6 living in poverty), so \(q=1 - p=1 - 0.23 = 0.77\)
• Margin of error \(E = 0.01\)
• For a \(99\%\) confidence level, \(\alpha=1 - 0.99 = 0.01\), and \(\alpha/2=0.005\). From the standard normal distribution table, \(z_{\alpha/2}=z_{0.005} = 2.576\)

Step2: Use the formula for sample size \(n\)

The formula for sample size when estimating a proportion is \(n=\frac{z_{\alpha/2}^{2}\times p\times q}{E^{2}}\)
Substitute the values:

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Answer:

\(11751\)