Question

U.S. Postal Service’s performance. The U.S. Postal Service (USPS) reports that 95% of first-class mail within the same city is delivered on time (i.e., within 2 days of the time of mailing). To gauge the USPS performance, Price Waterhouse monitored the delivery of first-class mail items between Dec. 10 and Mar. 3—the most difficult delivery season due to bad weather conditions and holidays. In a sample of 332,000 items, Price Waterhouse determined that 282,200 were delivered on time. Comment on the performance of USPS first-class mail service over this time period.

Short answer

The confidence interval is from 0.849 to 0.851.

Hence, the performance of U.S. Postal Service (USPS) first-class mail service over the time period is below the standard during this time period.

Step-by-step solution

Given information

The U.S Postal Service (USPS) reports that 95% of first-class mail within the same city is delivered on time (i.e., within 2 days of the time of mailing).

In a sample of 332000 items, Price Waterhouse determined that 282200 were delivered on time.

Finding the 95% confidence interval

The sample proportion is the point estimator of the population proportion p.

Computing the sample proportion is,

$\begin{array}{rcl}\hat{\mathrm{p}}& =& \frac{\mathrm{X}}{\mathrm{n}}\\ & =& \frac{282200}{332000}\\ & =& 0.85\end{array}$

Then the level of$100\left(1-\mathrm{\alpha }\right)\%$ confidence interval for p (proportion) is,

$\hat{\mathrm{p}}\pm {\mathrm{z}}_{\frac{\mathrm{\alpha }}{2}}\sqrt{\frac{\hat{\mathrm{p}}\left(1-\hat{\mathrm{p}}\right)}{\mathrm{n}}}$

For a 95% confidence interval, the value of$\frac{\mathrm{\alpha }}{2}$ is,

$\begin{array}{rcl}100(1-\mathrm{\alpha })\%& =& 95\%\\ (1-\mathrm{\alpha })& =& 0.95\end{array}$

For,$\mathrm{\alpha }=0.05\mathrm{and}\frac{\mathrm{\alpha }}{2}=0.025$

The 95% confidence interval is,

$\begin{array}{rcl}\hat{\mathrm{p}}\pm {\mathrm{z}}_{\frac{\mathrm{\alpha }}{2}}\sqrt{\frac{\hat{\mathrm{p}}\left(1-\hat{\mathrm{p}}\right)}{\mathrm{n}}}& =& 0.85\pm 1.960\sqrt{\frac{0.85(1-0.85)}{332000}}\left[\mathrm{From}\mathrm{Standard}\mathrm{Normal}\mathrm{Table}\right]\\ & =& \left(0.85\pm 1.960\sqrt{0.0006197}\right)\\ & =& \left(0.85\pm 0.001215\right)\\ & =& \left(0.848785,0.851215\right)\end{array}$

So, 95% confidence interval that the true population of first-class mail within the same city that is delivered between 0.849 and 0.851, i.e., 84.9% and 85.1%

This interval does not contain the reported 95% of the first-classmail delivered on time.

Hence, the performance of U.S. Postal Service (USPS) first-class mail service over the time period is below the standard during this time period.