# Express the confidence interval 53.5\%<p<69.7\% in the form of \widehat{p}\pm

Express the confidence interval $$\displaystyle{53.5}\%{<}{p}{<}{69.7}\%$$ in the form of $$\displaystyle\hat{{{p}}}\pm{M}{E}\hat{{{p}}}\pm{M}{E}$$

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Step 1
Solution
Given:
Confidence interval in the trilinear inequality form:
Lower limit $$\displaystyle{<}{p}{<}$$ Upper limit
$$\displaystyle{53.5}\%{<}{p}{<}{69.7}\%$$
Hence,
Lower limit $$\displaystyle={53.5}\%$$
Upper limit $$\displaystyle={69.7}\%$$
Step 2
Sample proportion $$\displaystyle{\left(\hat{{{p}}}\right)}={\frac{{\text{Upper limit+Lower limit}}}{{{2}}}}$$
Plug in all the values in the formula, we get
Simple proportion $$\displaystyle{\left(\hat{{{p}}}\right)}={\frac{{{69.7}\%+{53.5}\%}}{{{2}}}}$$
Sample proportion $$\displaystyle{\left(\hat{{{p}}}\right)}={61.6}\%$$
Step 3
Margin of error $$\displaystyle{\left({M}{E}\right)}={\frac{{\text{Upper limit+Lower limit}}}{{{2}}}}$$
Plug in all the values in the formula, we get
Margin of error $$\displaystyle{\left({M}{E}\right)}={\frac{{{69.7}\%+{53.5}\%}}{{{2}}}}$$
Margin of error $$\displaystyle{\left({M}{E}\right)}={8.1}\%$$
Step 4
Express the confidence interval in the form $$\displaystyle{\left(\hat{{{p}}}\right)}\pm{E}$$
$$\displaystyle{\left(\hat{{{p}}}\right)}\pm{E}-{61.6}\%\pm{8.1}\%$$