# With 80% confidence, for sample proportion 0.46 and sample size 27, what is the upper confidence...

## Question:

With {eq}80\% {/eq} confidence, for sample proportion {eq}0.46 {/eq} and sample size {eq}27 {/eq}, what is the upper confidence limit with 2 decimal places?

## Confidence Interval for Population Proportion:

Confidence interval gives range of all values likely to be true population proportion at given level of confidence. The range of values is determined by sample size, sample variability and confidence interval.

Given that;

{eq}\hat p=0.46\\n=27 {/eq}

Use equation below to construct upper confidence bound (UCB) of the population proportion:

{eq}\displaystyle UCB=\hat p+z_{\frac{\alpha}{2}}\cdot \sqrt{\frac{\hat p(1-\hat p)}{n}} {/eq}

Find critical value that correspond to 80% level of confidence:

{eq}\displaystyle \frac{\alpha}{2}=\frac{1-0.80}{2}=0.1\\z_{0.1}=1.28 {/eq}

Calculate the UCB:

{eq}\begin{align*} \displaystyle UCB&=0.46+1.28\times \sqrt{\frac{0.46(1-0.46)}{27}}\\&=0.46+0.12\\&=0.58 \end{align*} {/eq}