# We know the total population is 3,000. 30% of the 500 people sampled say that they prefer hot...

## Question:

We know the total population is 3,000. 30% of the 500 people sampled say that they prefer hot dogs to hamburgers.

Find the 95% confidence interval for the proportion that prefer hot dogs.

## Confidence Interval

A confidence interval is an interval of values estimated using the sample statistics obtained from a population, which are likely to contain the true value of our population parameter, with a certain degree of confidence. The confidence interval depends on factors like sample size, standard deviation and mean of the data.

The formula for confidence interval for proportion is:

{eq}\hat p \pm {z_{1 - \dfrac{\alpha }{2}}} \cdot \sqrt {\dfrac{{\hat p\left( {1 - \hat p} \right)}}{n}} {/eq}.

Given Information:

Of n = 500 selected people, 0.3 percent liked hot dogs.

The sample proportion is:

{eq}\hat p = 0.3 {/eq}.

The lower limit of 95% confidence interval for the proportion is:

{eq}\begin{align*} \hat p - {z_{0.975}} \cdot \sqrt {\dfrac{{\hat p\left( {1 - \hat p} \right)}}{n}} &= 0.3 - \left( {1.96} \right) \cdot \sqrt {\dfrac{{0.3\left( {1 - 0.3} \right)}}{{500}}} \\ &= 0.259832 \end{align*} {/eq}.

The upper limit of 95% confidence interval for the proportion is:

{eq}\begin{align*} \hat p + {z_{0.975}} \cdot \sqrt {\dfrac{{\hat p\left( {1 - \hat p} \right)}}{n}} &= 0.3 + \left( {1.96} \right) \cdot \sqrt {\dfrac{{0.3\left( {1 - 0.3} \right)}}{{500}}} \\ &= 0.340168 \end{align*} {/eq}

The z-value is obtained using the z-tables.

Therefore, the 95% confidence interval for the true proportion of the people who like hot dogs are {eq}\left( {0.259832,0.340168} \right). {/eq} 