Skip to main content
Contents Embed Dark Mode Prev Up Next \(\newcommand{\N}{\mathbb N} \newcommand{\Z}{\mathbb Z} \newcommand{\Q}{\mathbb Q} \newcommand{\R}{\mathbb R}
\newcommand{\lt}{<}
\newcommand{\gt}{>}
\newcommand{\amp}{&}
\definecolor{fillinmathshade}{gray}{0.9}
\newcommand{\fillinmath}[1]{\mathchoice{\colorbox{fillinmathshade}{$\displaystyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\textstyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\scriptstyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\scriptscriptstyle\phantom{\,#1\,}$}}}
\)
Section 9.5 Confidence Intervals for Proportions
We can also estimate the proportion of a population by constructing a confidence interval from a sample. Recall that proportion data follow the binomial distribution, which can be approximated by the normal distribution under the conditions:
Figure 9.5.1.
Definition 9.5.2 .
The
confidence interval for the proportion is an interval estimate around a sample proportion that provides a range of where
\(p\) lies.
The formula for this confidence interval is
\begin{equation*}
\overline{p}\pm z_{\alpha/2}\cdot\hat{\sigma}_p,
\end{equation*}
where \(\overline{p}=\frac{x}{n}\) and \(\hat{\sigma}_p=\sqrt{\frac{\overline{p}(1-\overline{p})}{n}}\text{.}\)
The sample proportion,
\(\overline{p}\text{,}\) measures the fraction of “successes” in the sample.
Exercise 9.5.3 .
The IRS reported that
\(85\%\) of individual tax returns were filed electronically in 2017. A random sample of 240 tax returns from 2018 was selected. From this sample, 187 were filed electronically.
(a) What is the point estimate based on this sample?
Answer .
\begin{equation*}
\overline{p}=\frac{187}{240}\approx 0.779
\end{equation*}
(b) What is the margin of error for this sample?
Answer .
\(\hat{\sigma_p}=\sqrt{\frac{.779(1-.779)}{240}}\approx .0268\)
\(\alpha=0.10\rightarrow\alpha/2=.05\rightarrow z_{.05}=NORM.S.INV(.95)\approx 1.645\)
\(ME_{\overline{p}}\approx 1.645\cdot (.0268)\)
\begin{equation*}
ME_{\overline{p}}\approx 0.0441
\end{equation*}
(c) Construct a
\(90\%\) confidence interval to estimate the actual proportion of taxpayers who filed electronically in 2018?
Answer .
\(\overline{p}\pm ME_{\overline{p}}\rightarrow .779\pm .0441\)
We are
\(90\%\) confident that
\(0.7349\lt p \lt 0.8231\text{.}\) (That is, we are
\(90\%\) confident that the proportion of taxpayers who filed electronically in 2018 is between
\(73.49\%\) and
\(82.31\%\text{.}\) )
(d) Is there any evidence that this proportion has changed since 2017 based on this sample?
Answer .
Since 0.85 does not fall in the interval, there is evidence that the proportion of electronic filers has changed.
