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QUAN 2010 Notes: Introduction to Business Statistics

Omar Ortega

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:
  • \(np\geq 5\) and
  • \(n(1-p)\geq 5\text{,}\)
Figure 9.5.1.
Binomial and Normal Distributions \(z\)-value (Made in GeoGebra by Steve Phelps)

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.

(Donnelly 8.34)
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.
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