Question

Minority ownership of franchises. According to a 2011 report for IFA Educational Foundation, 20.5% of all franchised businesses in the United States are minority owned. (This information is based on the U.S. Census Bureau’s survey of 27 million business owners.) Suppose that you obtain a sample of 100 franchised businesses located in Mississippi and find that 15 are owned by minorities. Does this result lead you to conclude that the percentage of minority-owned franchises in Mississippi is less than the national value of 20.5%? Explain.

Short answer

The confidence interval is (0.88,0.22).

No,the result do no lead us to conclude that the percentage of minority-owned franchises in Mississippi is less than the national value of 20.5%.

Step-by-step solution

Given information

According to a 2011 report for IFA Educational Foundation, a sample of 100 franchised businesses located in Mississippi and find that 15 are owned by minorities.

Finding the 95% confidence interval

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

Computing the sample proportion is,

$\begin{array}{rcl}\hat{\mathrm{p}}& =& \frac{\mathrm{X}}{\mathrm{n}}\\ & =& \frac{15}{100}\\ & =& 0.15\end{array}$

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,

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

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

The 95% confidence interval is,

$\begin{array}{rcl}\hat{\mathrm{p}}\pm {\mathrm{z}}_{\frac{\mathrm{\alpha }}{2}}\sqrt{\frac{\hat{\mathrm{p}}\left(1-\hat{\mathrm{p}}\right)}{\mathrm{n}}}& =& 0.15\pm 1.960\sqrt{\frac{0.15\left(1-0.15\right)}{100}}\left[\mathrm{From}\mathrm{Standard}\mathrm{Normal}\mathrm{Table}\right]\\ & =& \left(0.15\pm 1.960\sqrt{0.001275}\right)\\ & =& \left(0.15\pm 0.07\right)\\ & =& \left(0.08,0.02\right)\end{array}$

Therefore, the 95% confidence interval that the true percentage of minority-owned franchises in Mississippi is between 8% and 22%.

Also 20.5% falls in this interval.

Hence, conclude that the percentage of minority-owned franchises in Mississippi is not less than the national value.