document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1198867>Question 1198867</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Of 41 bank customers depositing a check, 20 received some cash back.\r<BR>\n" );
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document.write( "(a) Construct a 90 percent confidence interval for the proportion of all depositors who ask for cash back </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #848402 by </B><A HREF=/tutors/aboutme.mpl?userid=ElectricPavlov><B>ElectricPavlov(122)</B></A><img src=\"/images/stars/stars-09.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=ElectricPavlov><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=848402\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> **1. Find the sample proportion (p̂)**\r<BR>\n" );
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document.write( "* p̂ = (Number of depositors who received cash back) / (Total number of depositors)<BR>\n" );
document.write( "* p̂ = 20 / 41 <BR>\n" );
document.write( "* p̂ ≈ 0.4878\r<BR>\n" );
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document.write( "**2. Find the standard error (SE)**\r<BR>\n" );
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document.write( "* SE = √(p̂ * (1 - p̂) / n) <BR>\n" );
document.write( "* SE = √(0.4878 * (1 - 0.4878) / 41) <BR>\n" );
document.write( "* SE ≈ 0.0764\r<BR>\n" );
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document.write( "**3. Find the critical value (z*) for a 90% confidence level**\r<BR>\n" );
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document.write( "* For a 90% confidence level, the area in each tail of the standard normal distribution is (100% - 90%) / 2 = 5%.<BR>\n" );
document.write( "* Using a standard normal distribution table or calculator, the critical value (z*) for 5% in the tail is 1.645.\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 = z* * SE <BR>\n" );
document.write( "* Margin of Error = 1.645 * 0.0764 <BR>\n" );
document.write( "* Margin of Error ≈ 0.1257\r<BR>\n" );
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document.write( "**5. Construct the 90% confidence interval**\r<BR>\n" );
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document.write( "* Lower limit = p̂ - Margin of Error = 0.4878 - 0.1257 = 0.3621<BR>\n" );
document.write( "* Upper limit = p̂ + Margin of Error = 0.4878 + 0.1257 = 0.6135\r<BR>\n" );
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document.write( "**Therefore, the 90% confidence interval for the proportion of all depositors who ask for cash back is approximately (0.3621, 0.6135).**\r<BR>\n" );
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document.write( "This means we are 90% confident that the true proportion of all depositors who ask for cash back lies between 36.21% and 61.35%.<BR>\n" );
document.write( " </SPAN>\n" );
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