Question

if n = 470 and $widehat{p}$ (p - hat) = 0.4, 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. < p < question help: video message instructor post to forum submit question

Explanation:

Step1: Calculate number of successes

$x = n\times\hat{p}=470\times0.4 = 188$

Step2: Identify z - value for 99% CI

For a 99% confidence interval, the z - value $z_{\alpha/2}= 2.576$

Step3: Calculate standard error

The standard error $SE=\sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}=\sqrt{\frac{0.4\times(1 - 0.4)}{470}}=\sqrt{\frac{0.4\times0.6}{470}}\approx\sqrt{\frac{0.24}{470}}\approx0.0226$

Step4: Calculate lower bound

The lower bound of the confidence interval is $\hat{p}-z_{\alpha/2}\times SE=0.4 - 2.576\times0.0226=0.4 - 0.0582\approx0.342$

Step5: Calculate upper bound

The upper bound of the confidence interval is $\hat{p}+z_{\alpha/2}\times SE=0.4 + 2.576\times0.0226=0.4+ 0.0582\approx0.458$

Answer:

$0.342 < p < 0.458$