document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1194768>Question 1194768</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A researcher wishes to estimate the proportion of adults who have​ high-speed Internet access. What size sample should be obtained if she wishes the estimate to be within 0.03 with ​90% confidence if<BR>\n" );
document.write( "​(a) she uses a previous estimate of ​0.32?<BR>\n" );
document.write( "​(b) she does not use any prior​ estimates?<BR>\n" );
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document.write( "Part (a)\r<BR>\n" );
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document.write( "Use this table<BR>\n" );
document.write( "<a  rel=nofollow HREF= \"https://www.sjsu.edu/faculty/gerstman/StatPrimer/t-table.pdf\">https://www.sjsu.edu/faculty/gerstman/StatPrimer/t-table.pdf</a><BR>\n" );
document.write( "to find the z critical value at 90% confidence is approximately z = 1.645\r<BR>\n" );
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document.write( "p = 0.32 is the previous estimate of the proportion<BR>\n" );
document.write( "E = 0.03 is the error we want (or smaller)\r<BR>\n" );
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document.write( "n = sample size<BR>\n" );
document.write( "n = p*(1-p)*(z/E)^2<BR>\n" );
document.write( "n = 0.32*(1-0.32)*(1.645/0.03)^2<BR>\n" );
document.write( "n = 654.256711111111<BR>\n" );
document.write( "<font color=red>n = 655</font>\r<BR>\n" );
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document.write( "We always round UP to the nearest whole number when doing min sample size problems. <BR>\n" );
document.write( "It doesn't matter that 654.256711111111 is closer to 654 than it is to 655\r<BR>\n" );
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document.write( "We round up to clear the hurdle needed. <BR>\n" );
document.write( "Try n = 654 in the margin of error formula<BR>\n" );
document.write( "E = z*sqrt(p*(1-p)/n)<BR>\n" );
document.write( "and you'll find that E > 0.03 which isn't what we want.<BR>\n" );
document.write( "But if you tried n = 655, then E = 0.03 or E < 0.03 would be the case.\r<BR>\n" );
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document.write( "<font color=red>Answer: 655</font>\r<BR>\n" );
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document.write( "Part (b)\r<BR>\n" );
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document.write( "This is going to be very similar to part (a). <BR>\n" );
document.write( "However, we'll be using p = 0.50 this time. <BR>\n" );
document.write( "This is the most conservative estimate or guess to make for p if we don't know what it is. <BR>\n" );
document.write( "It's right in the middle of the interval <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0+%3C=+p+%3C=+1 ALIGN=MIDDLE  ALT=\"0+%3C=+p+%3C=+1\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "The other values of z = 1.645 and E = 0.03 remain the same from before.\r<BR>\n" );
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document.write( "n = sample size<BR>\n" );
document.write( "n = p*(1-p)*(z/E)^2<BR>\n" );
document.write( "n = 0.50*(1-0.50)*(1.645/0.03)^2<BR>\n" );
document.write( "n = 751.673611111111<BR>\n" );
document.write( "<font color=red>n = 752</font><BR>\n" );
document.write( "Once again, always round up.\r<BR>\n" );
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document.write( "Since p = 0.5, this means p(1-p) = 0.5*(1-0.5) = 0.5*0.5 = 0.25<BR>\n" );
document.write( "The formula above updates to n = 0.25*(z/E)^2 when using p = 0.5\r<BR>\n" );
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document.write( "<font color=red>Answer: 752</font>\r<BR>\n" );
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document.write( "Further Reading:<BR>\n" );
document.write( "<a  rel=nofollow HREF= \"http://www.ltcconline.net/greenl/courses/201/estimation/ciprop.htm\">http://www.ltcconline.net/greenl/courses/201/estimation/ciprop.htm</a><BR>\n" );
document.write( "Scroll down to the section with the subheading of \"Finding n to Estimate a Proportion\". <BR>\n" );
document.write( "I have no affiliation to the websites in either link mentioned.<BR>\n" );
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