A federal report stated that 88% of children under 18 were covered by health insurance in 2000. How large a sample is needed to estimate the true proportion of covered children with 90% confidence with a confidence interval of .05 wide?

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A federal report stated that 88% of children under 18 were covered by health insurance in 2000. How large a sample is needed to estimate the true proportion of covered children with 90% confidence with a confidence interval of .05 wide?

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Answer 1 · 2018-04-15T00:53:56

$n = 115$ $n = 115$

Explanation:

Do you mean with a margin of error of $5 %$ $5%$ ?

The formula for a confidence interval for a proportion is given by $ˆ p \pm M E$ $hat p +- ME$ , where $M E = z$ $ME = z$ * $\cdot S E (ˆ p)$ $* SE (hat p)$ .

$ˆ p$ $hat p$ is the sample proportion

$z$ $z$ * is the critical value of $z$ $z$ , which you can obtain from a graphing calculator or a table

$S E (ˆ p)$ $SE(hat p)$ is the standard error of the sample proportion, which can be found using $\sqrt{\frac{ˆ p ˆ q}{n}}$ $sqrt ((hat p hat q)/n)$ , where $ˆ q = 1 - ˆ p$ $hat q = 1 - hat p$ and $n$ $n$ is the sample size

We know that the margin of error should be $0.05$ $0.05$ . With a $90 %$ $90%$ confidence interval, $z$ $z$ * $\approx 1.64$ $~~ 1.64$ .

$M E = z$ $ME = z$ * $\cdot S E (ˆ p)$ $* SE (hat p)$

$0.05 = 1.64 \cdot \sqrt{\frac{0.88 \cdot 0.12}{n}}$ $0.05 = 1.64 * sqrt ((0.88 * 0.12)/n)$

We can now solve for $n$ $n$ algebraically. We get $n \approx 114.2$ $n ~~ 114.2$ , which we round up to $115$ $115$ because a sample size of $114$ $114$ would be too small.

We need at least $115$ $115$ children to estimate the true proportion of children who are covered by health insurance with $90 %$ $90%$ confidence and a margin of error of $5 %$ $5%$ .

Answer 2 · 2018-04-15T05:06:20

458

Explanation:

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A federal report stated that 88% of children under 18 were covered by health insurance in 2000. How large a sample is needed to estimate the true proportion of covered children with 90% confidence with a confidence interval of .05 wide?

2 Answers

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