A random sample of size n = 225 yielded $\hat{p}$= .46

a. Is the sample size large enough to use the methods of this section to construct a confidence interval for p? Explain.

b. Construct a 95% confidence interval for p.

c. Interpret the 95% confidence interval.

d. Explain what is meant by the phrase “95% confidence interval.”

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$n\hat{p}⩾15,n\left(1-\hat{p}\right)⩾15$

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

Hence the sample is large enough

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

The margin of error

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

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 “95% confidence interval” means that the 95% interval gives us an interval that contains the population parameter with 95% confidence.