document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1139536>Question 1139536</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Can someone please help my son with solving this question:\r<BR>\n" );
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document.write( "In a survey of 1051 ​adults, a poll​ asked, \"Are you worried or not worried about having enough money for​ retirement?\" Of the 1051 ​surveyed, 599 stated that they were worried about having enough money for retirement. Construct a 95​% confidence interval for the proportion of adults who are worried about having enough money for retirement.\r<BR>\n" );
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document.write( "A 95​% confidence interval for the proportion of adults who are worried about having enough money for retirement is  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #760014 by </B><A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><B>jim_thompson5910(35256)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><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=760014\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font face=\"times\" color=\"black\" size=\"3\"><BR>\n" );
document.write( "n = 1051 is the sample size\r<BR>\n" );
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document.write( "Of that sample size, x = 599 said they were worried, so <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \hat{p} = \frac{x}{n} = \frac{599}{1051} \approx 0.569933\"><BR>\n" );
document.write( "is a good estimate of the population proportion of people worried about having enough money for retirement.\r<BR>\n" );
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document.write( "The notation <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \hat{p}\"> is read as \"p-hat\". Basically it's the letter \"p\" but with a \"hat\" on top so to speak. This is to help separate it from the regular letter \"p\" which is the population proportion; while <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \hat{p}\"> is the sample proportion.\r<BR>\n" );
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document.write( "Let's compute the standard error which I'll call \"SE\" for short.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \text{SE} = \sqrt{\frac{\hat{p}*(1-\hat{p})}{n}}\"> see note below\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \text{SE} \approx \sqrt{\frac{0.569933*(1-0.569933)}{1051}}\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \text{SE} \approx \sqrt{\frac{0.569933*0.430067}{1051}}\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \text{SE} \approx \sqrt{\frac{0.245109}{1051}}\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \text{SE} \approx 0.015264\">\r<BR>\n" );
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document.write( "note: we do not use p here as we don't know the population proportion. If we knew the population proportion then we wouldn't need a confidence interval (since a confidence interval is used to estimate the population proportion). We can say that p-hat is an unbiased estimator of p.\r<BR>\n" );
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document.write( "Now onto the margin of error, which I'll abbreviate as \"ME\". We'll need the z critical value. At 95% confidence, the critical z value is approximately z = 1.960; this value is found using a calculator or a table. \r<BR>\n" );
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document.write( "I used <a  rel=nofollow HREF= \"http://www.sjsu.edu/faculty/gerstman/StatPrimer/t-table.pdf\">this table</a> to find the z critical value. Scroll to the bottom of the page to locate the row that starts with Z. Then locate the column that has \"95%\" at the very bottom. The value just above this is 1.960. A table similar to this should be found in the appendix section of your statistics textbook.\r<BR>\n" );
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document.write( "Let's use those two values to get...\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \text{ME} = z*\text{SE}\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \text{ME} \approx 1.960*0.015264\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \text{ME} \approx 0.029917\">\r<BR>\n" );
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document.write( "The margin of error is then added and subtracted from the p-hat value, as the p-hat value is the best estimate of p. The p-hat value is the center of the confidence interval. The margin of error tells us how spread out or how wide the interval is.\r<BR>\n" );
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document.write( "L = lower boundary of confidence interval\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large L = \hat{p} - \text{ME}\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large L \approx 0.569933 - 0.029917\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large L \approx 0.540016\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large L \approx 0.54\">\r<BR>\n" );
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document.write( "The upper boundary is nearly identical, but instead we add this time.\r<BR>\n" );
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document.write( "U = upper boundary of confidence interval\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large U = \hat{p} + \text{ME}\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large U \approx 0.569933 + 0.029917\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large U \approx 0.59985\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large U \approx 0.60\">\r<BR>\n" );
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document.write( "Answer: <font color=red size=4>(0.54, 0.60)</font><BR>\n" );
document.write( "This answer is approximate rounded to two decimal places. \r<BR>\n" );
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document.write( "Interpretation: We are 95% confident that the true proportion p is between 0.54 and 0.60, meaning that we're 95% confident that the proportion of people worried about having enough money for retirement is between 54% and 60%.<BR>\n" );
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