document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1204948>Question 1204948</A>: <I><SPAN STYLE=\"word-break: break-all;\"> In a recent year, the Better Business Bureau settled 75% of complaints they received. You have been hired by the Bureau to investigate complaints this year involving computer stores. Assume the population proportion of complaints settled for the computer stores is the 0.75. Suppose your sample size is 152. What is the probability that the sample proportion will be within 4 percentage points of the population proportion?\r<BR>\n" );
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document.write( "my work: <BR>\n" );
document.write( ".75=x 152=n<BR>\n" );
document.write( ".04*.75=.03=Z<BR>\n" );
document.write( "x/sqrt n = .75/sqrt 152 = 0.7097<BR>\n" );
document.write( ".75+.03=.78<BR>\n" );
document.write( ".75-.03=.72<BR>\n" );
document.write( "(.78-.75)/0.7097=0.0423<BR>\n" );
document.write( "(.72-.75)/0.7097=-0.0423<BR>\n" );
document.write( "difference = 0.0337 \r<BR>\n" );
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document.write( "the issue is that I know I have done something wrong because the answer key says that I am supposed to have gotten 0.7456 and I do not understand what I am doing wrong or which steps that I have mixed up or missed so that I am getting something so different.  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #841502 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3817)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><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=841502\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "p = population proportion<BR>\n" );
document.write( "phat = sample proportion<BR>\n" );
document.write( "n = sample size\r<BR>\n" );
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document.write( "The claim is that p = 0.75<BR>\n" );
document.write( "4 percentage points within this value represents the interval from 0.71 to 0.79 (because 0.75-0.04 = 0.71 and 0.75+0.04 = 0.79)\r<BR>\n" );
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document.write( "We're tasked to compute P(0.71 < phat < 0.79)\r<BR>\n" );
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document.write( "When phat = 0.71, we get this z score<BR>\n" );
document.write( "z = (phat - p)/( sqrt(p*(1-p)/n) )<BR>\n" );
document.write( "z = (0.71 - 0.75)/( sqrt(0.75*(1-0.75)/152) )<BR>\n" );
document.write( "z = -1.138888<BR>\n" );
document.write( "z = -1.14\r<BR>\n" );
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document.write( "When phat = 0.79, we get this z score<BR>\n" );
document.write( "z = (phat - p)/( sqrt(p*(1-p)/n) )<BR>\n" );
document.write( "z = (0.79 - 0.75)/( sqrt(0.75*(1-0.75)/152) )<BR>\n" );
document.write( "z = 1.138888<BR>\n" );
document.write( "z = 1.14\r<BR>\n" );
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document.write( "Side note: sqrt(p*(1-p)/n) represents the standard error (SE)<BR>\n" );
document.write( "refer to page 2 of this formula sheet<BR>\n" );
document.write( "<a  rel=nofollow HREF=\"https://apcentral.collegeboard.org/media/pdf/statistics-formula-sheet-and-tables-2020.pdf\">https://apcentral.collegeboard.org/media/pdf/statistics-formula-sheet-and-tables-2020.pdf</a>\r<BR>\n" );
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document.write( "P(0.71 < phat < 0.79) is roughly equivalent to P(-1.14 < z < 1.14) when p = 0.75 and n = 152.\r<BR>\n" );
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document.write( "Next I'll use a Z table as shown here<BR>\n" );
document.write( "<a  rel=nofollow HREF=\"https://www.ztable.net/\">https://www.ztable.net/</a><BR>\n" );
document.write( "This table should be found in the back of your stats textbook. <BR>\n" );
document.write( "On exams your teacher will likely hand out the table. \r<BR>\n" );
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document.write( "In the table, locate the row that starts with -1.1<BR>\n" );
document.write( "Place a piece of paper under it so you keep track of this row. <BR>\n" );
document.write( "If working a digital copy, then open your favorite paint program to mark this row.<BR>\n" );
document.write( "Next, highlight the column that has 0.04 at the top.<BR>\n" );
document.write( "The intersection of this row and column yields the value 0.12714<BR>\n" );
document.write( "It means P(z < -1.14) = 0.12714 approximately<BR>\n" );
document.write( "The area under the standard normal Z curve to the left of -1.14 is roughly 0.12714\r<BR>\n" );
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document.write( "Follow similar steps to find that P(z < 1.14) = 0.87286\r<BR>\n" );
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document.write( "Then we subtract the items to get the area between<BR>\n" );
document.write( "P(a < Z < b) = P(Z < b) - P(Z < a)<BR>\n" );
document.write( "P(-1.14 < Z < 1.14) = P(Z < 1.14) - P(Z < -1.14)<BR>\n" );
document.write( "P(-1.14 < Z < 1.14) = 0.87286 - 0.12714<BR>\n" );
document.write( "P(-1.14 < Z < 1.14) = 0.74572\r<BR>\n" );
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document.write( "This leads back to P(0.71 < phat < 0.79) = 0.74572 which rounds to 0.7457\r<BR>\n" );
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document.write( "The discrepancy of 0.7457 and 0.7456 is likely due to some rounding error.<BR>\n" );
document.write( "It's possible that I may have made a mistake somewhere. <BR>\n" );
document.write( "Perhaps your table uses slightly different values.\r<BR>\n" );
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document.write( "--------------------------\r<BR>\n" );
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document.write( "If you want to use a calculator then you have plenty of options to pick from.\r<BR>\n" );
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document.write( "On TI83 and TI84 calculators, use the normalCDF function<BR>\n" );
document.write( "<a  rel=nofollow HREF=\"https://www.statology.org/normal-probabilities-ti-84-calculator/\">https://www.statology.org/normal-probabilities-ti-84-calculator/</a>\r<BR>\n" );
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document.write( "If your TI calculator input is <font color=red>normalCDF(-1.1389,1.1389,0,1)</font> then the result is approximately 0.745 255 0333 which rounds to 0.7453\r<BR>\n" );
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document.write( "If your TI calculator input is <font color=red>normalCDF(-1.138888,1.138888,0,1)</font> then the result is approximately 0.745 250 0276 which rounds to 0.7453\r<BR>\n" );
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document.write( "Play around with the accuracy of the 1.138888 values to see how the area changes. The change shouldn't be too much.\r<BR>\n" );
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document.write( "Another calculator to use is this online one<BR>\n" );
document.write( "<a  rel=nofollow HREF=\"https://davidmlane.com/hyperstat/z_table.html\">https://davidmlane.com/hyperstat/z_table.html</a>\r<BR>\n" );
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document.write( "Yet another option is to use the spreadsheet function <font color=red>NormDist</font><BR>\n" );
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