91Ó°ÊÓ

The given data is about a survey conducted to know the music preferences of the citizens

The objective is to find the confidence interval and error bound of the data

A poll of music tastes was recently done by Public Policy Polling. When asked, $x=80$of the $n=571$participants acknowledged to downloading music illegally.

As a result, the sample's estimated proportion is

$p^{\text{'}}=\frac{x}{n}=\frac{80}{571}=0.14.$

If the proportion of observations in a random sample of size $n$ that belong to a class of interest is $p^{\text{'}}$, an approximate $100(1-\mathrm{Î\pm })\%$ confidence interval on the proportion $p$ of the population that belongs to this class is $p^{\text{'}}$.

where $z_{\frac{\mathrm{Î\pm }}{2}}$is the upper $\frac{\mathrm{Î\pm }}{2}$percentage point of the standard normal distribution. For $99\%$ two-sided confidence interval,

and

From the Equations, a $95\%$ two-sided $CL$ for the population proportion is

Therefore, a $99\%$ $CL$ for $p$ is

b) Many people will not respond to telephone interviews surveys. If they do respond to the surveys, you can not be sure who is responding.

c) Decreasing the confidence level causes the error bound to decrease, making the confidence interval narrower.