document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1197733>Question 1197733</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A dentist is interested in the proportion of adults who have had their wisdom teeth removed. To determine this, a random sample of {227} adults was collected and it was found that {105} still had their wisdom teeth. \r<BR>\n" );
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document.write( "Using this information construct an approximate 90% confidence interval for the proportion of adults who have had their wisdom teeth removed. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #848489 by </B><A HREF=/tutors/aboutme.mpl?userid=proyaop><B>proyaop(69)</B></A><img src=\"/images/stars/stars-08.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=proyaop><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=848489\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> **1. Calculate the Sample Proportion (p̂)**\r<BR>\n" );
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document.write( "* p̂ = (Number of adults who have had wisdom teeth removed) / (Sample size)<BR>\n" );
document.write( "* p̂ = (227 - 105) / 227 <BR>\n" );
document.write( "* p̂ = 122 / 227 <BR>\n" );
document.write( "* p̂ ≈ 0.537 \r<BR>\n" );
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document.write( "**2. Find the Critical Value (z*)**\r<BR>\n" );
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document.write( "* For a 90% confidence interval, the critical value (z*) is 1.645. This value corresponds to the z-score that captures 90% of the data within two standard deviations of the mean in a standard normal distribution.\r<BR>\n" );
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document.write( "**3. Calculate the Standard Error**\r<BR>\n" );
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document.write( "* Standard Error (SE) = √(p̂ * (1 - p̂) / n) <BR>\n" );
document.write( "    * where:<BR>\n" );
document.write( "        * p̂ = sample proportion (0.537)<BR>\n" );
document.write( "        * n = sample size (227)\r<BR>\n" );
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document.write( "* SE = √(0.537 * (1 - 0.537) / 227) <BR>\n" );
document.write( "* SE ≈ 0.033\r<BR>\n" );
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document.write( "**4. Calculate the Margin of Error**\r<BR>\n" );
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document.write( "* Margin of Error (ME) = z* * SE <BR>\n" );
document.write( "* ME = 1.645 * 0.033 <BR>\n" );
document.write( "* ME ≈ 0.054\r<BR>\n" );
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document.write( "**5. Construct the Confidence Interval**\r<BR>\n" );
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document.write( "* Lower Limit = p̂ - ME = 0.537 - 0.054 = 0.483<BR>\n" );
document.write( "* Upper Limit = p̂ + ME = 0.537 + 0.054 = 0.591\r<BR>\n" );
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document.write( "**Therefore, the 90% confidence interval for the proportion of adults who have had their wisdom teeth removed is (0.483, 0.591).**\r<BR>\n" );
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document.write( "**Interpretation:**\r<BR>\n" );
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document.write( "We are 90% confident that the true proportion of adults who have had their wisdom teeth removed lies between 48.3% and 59.1%.<BR>\n" );
document.write( " </SPAN>\n" );
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