91影视

A random sample of size n = 225 yielded $\hat{p}$= .46. It is required to compute or explain the following

a. If the sample size is large enough to construct a 95% confidence interval

, b.95% confidence interval for the population proportion p

c. Interpretation of 95% confidence interval

d. meaning of the phrase 95% confidence interval.

We know that the sample size is large enough only if$n\hat{p}猢�15,n\left(1-\hat{p}\right)猢�15$

Here,$n\hat{p}=225脳0.46=103.5猢�15$ and$n\left(1-\hat{p}\right)=225脳\left(1-0.46\right)=121.5猢�15$

Hence the sample is large enough

Here the confidence coefficient is 95%, hence $伪=5\%$Now from the standard normal distribution table,$z_{\frac{伪}{2}}=1.96$

The margin of error

$\begin{array}{rcl}E& =& z_{\frac{伪}{2}}脳\sqrt{\frac{\hat{p}\left(1-\hat{p}\right)}{n}}\\ & =& 1.96脳\sqrt{\frac{0.46脳\left(1-0.46\right)}{225}}\\ & =& 0.0651\\ & & \end{array}$

Hence 95% lower limit:$\hat{p}-E=0.46-0.0651=0.3949$

95% upper limit:$\hat{p}+E=0.46+0.0651=0.5251$

Hence 95% confidence interval for p is$\left(0.3949,0.5251\right)$

One is 95% confident that the true proportion lies between the interval $\left(0.3949,0.5251\right)$.

The phrase 鈥�95% confidence interval鈥� means that the 95% interval gives us an interval that contains the population parameter with 95% confidence.