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.

Solution

VerifiedGiven:

$\begin{aligned} n&=\text{Sample size}=100 \\ x&=\text{Number of successes}=15 \\ c&=\text{Confidence coefficient}=95\%=0.95 &\textcolor{#4559AC}{\text{(Assumption)}} \end{aligned}$

We need to determine a confidence interval about the population proportion $p$.

The confidence interval about the population proportion is $\hat{p}\pm E$ with $E=z_{\alpha/2}\cdot \sqrt{\dfrac{\hat{p}\hat{q}}{n}}$.

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