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

Efan Halliday 2021-09-23 Answered

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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Expert Answer

faldduE
Answered 2021-09-24 Author has 23536 answers

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 \(\displaystyle{\left(\hat{{{p}}}\right)}\pm{E}\)
\(\displaystyle{\left(\hat{{{p}}}\right)}\pm{E}-{61.6}\%\pm{8.1}\%\)

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