Question

8.2 one - sample interval for the proportion. construct and interpret a confidence interval for the population proportion.
if n = 520 and $widehat{p}$ (p - hat) = 0.3, construct a 99% confidence interval.
in order to use technology, you need to compute the number of successes:
x = n*$widehat{p}$
give your answers to three decimals.
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Explanation:

Step1: Calculate number of successes

$x = n\times\widehat{p}=520\times0.3 = 156$

Step2: Determine z - value for 99% confidence level

For a 99% confidence interval, the significance level $\alpha=1 - 0.99 = 0.01$, and $\alpha/2=0.005$. The $z$-value $z_{\alpha/2}=z_{0.005} = 2.576$

Step3: Calculate the margin of error E

The formula for the margin of error for a proportion is $E = z_{\alpha/2}\sqrt{\frac{\widehat{p}(1 - \widehat{p})}{n}}$. Substitute $\widehat{p}=0.3$, $n = 520$ and $z_{\alpha/2}=2.576$ into the formula.
$E=2.576\sqrt{\frac{0.3\times(1 - 0.3)}{520}}=2.576\sqrt{\frac{0.3\times0.7}{520}}=2.576\sqrt{\frac{0.21}{520}}\approx2.576\times0.020 = 0.052$

Step4: Construct the confidence interval

The confidence - interval for the population proportion $p$ is given by $\widehat{p}-ESubstitute $\widehat{p}=0.3$ and $E = 0.052$ into the formula: $0.3-0.052 < p<0.3 + 0.052$

Answer:

$0.248 < p<0.352$