document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1201801>Question 1201801</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Based on historical data, your manager believes that 28% of the company's orders come from first-time customers. A random sample of 151 orders will be used to estimate the proportion of first-time-customers. What is the probability that the sample proportion is between 0.22 and 0.43?\r<BR>\n" );
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document.write( "Note: You should carefully round any z-values you calculate to 4 decimal places to match wamap's approach and calculations. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #836306 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=836306\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "Answer: <font color=red size=4>0.9497</font> (approximate)\r<BR>\n" );
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document.write( "Explanation:\r<BR>\n" );
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document.write( "I'm not familiar with wamap, so I won't be much help there. But I can help with the rest.\r<BR>\n" );
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document.write( "Given info:<BR>\n" );
document.write( "\"<font color=blue>Based on historical data, your manager believes that 28% of the company's orders come from first-time customers</font>\"<BR>\n" );
document.write( "This leads to p = 0.28 as the population proportion. We convert from percent form to decimal form. Move the decimal 2 spots to the left.<BR>\n" );
document.write( "The value of p is always between 0 and 1 inclusive, ie <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( "More given info:<BR>\n" );
document.write( "\"<font color=blue>A random sample of 151 orders will be used to estimate the proportion of first-time-customers.</font>\"<BR>\n" );
document.write( "meaning<BR>\n" );
document.write( "n = 151 = sample size\r<BR>\n" );
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document.write( "Summary so far<BR>\n" );
document.write( "p = 0.28 = population proportion<BR>\n" );
document.write( "n = 151 = sample size\r<BR>\n" );
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document.write( "Now we look at the phat distribution.<BR>\n" );
document.write( "Sometimes it is spelled out as \"p-hat\"<BR>\n" );
document.write( "It's named as such because it is the letter p with a small hat on top.<BR>\n" );
document.write( "phat = <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \hat{p}\">\r<BR>\n" );
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document.write( "That font is probably a bit distorted, so here's a link that might offer a better image<BR>\n" );
document.write( "<a  rel=nofollow HREF= \"https://tex.stackexchange.com/questions/250466/vector-with-boldface-and-a-hat-on-top\">https://tex.stackexchange.com/questions/250466/vector-with-boldface-and-a-hat-on-top</a>\r<BR>\n" );
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document.write( "The phat is the sample proportion.<BR>\n" );
document.write( "phat's job is to estimate p.\r<BR>\n" );
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document.write( "The phat distribution has these 2 properties<ul><li>mean = p</li><li>standard deviation = standard error = sqrt(p*(1-p)/n)</li></ul>I'll use \"SE\" as the abbreviation of \"standard error\".\r<BR>\n" );
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document.write( "In this case, we have<ul><li>mean = 0.28</li><li>SE = sqrt(0.28*(1-0.28)/151) = 0.036539 approximately</li></ul>We can use a normal distribution with these two parameters.\r<BR>\n" );
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document.write( "The goal is to compute P(0.22 < phat < 0.43) <BR>\n" );
document.write( "It gives the probability the sample proportion phat is between 0.22 and 0.43 \r<BR>\n" );
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document.write( "Let's compute the z score when phat = 0.22<BR>\n" );
document.write( "z = (phat - p)/SE<BR>\n" );
document.write( "z = (0.22 - 0.28)/0.036539<BR>\n" );
document.write( "z = -1.64208106406853<BR>\n" );
document.write( "z = -1.6421\r<BR>\n" );
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document.write( "Repeat for phat = 0.43<BR>\n" );
document.write( "z = (phat - p)/SE<BR>\n" );
document.write( "z = (0.43 - 0.28)/0.036539<BR>\n" );
document.write( "z = 4.10520266017132<BR>\n" );
document.write( "z = 4.1052\r<BR>\n" );
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document.write( "The task of computing<BR>\n" );
document.write( "P(0.22 < phat < 0.43)<BR>\n" );
document.write( "is roughly equivalent to<BR>\n" );
document.write( "P(-1.6421 < z < 4.1052)\r<BR>\n" );
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document.write( "Now use a calculator such as this one<BR>\n" );
document.write( "<a  rel=nofollow HREF= \"https://davidmlane.com/normal.html\">https://davidmlane.com/normal.html</a><BR>\n" );
document.write( "or something like a TI83 or TI84\r<BR>\n" );
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document.write( "Or you could use a table like this<BR>\n" );
document.write( "<a  rel=nofollow HREF= \"https://www.ztable.net/\">https://www.ztable.net/</a><BR>\n" );
document.write( "The drawback with the table is you must round each z score to 2 decimal places. Thereby losing accuracy.\r<BR>\n" );
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document.write( "I'll go with the calculator approach. I'll assume the calculator you're using is similar to the one in the link.\r<BR>\n" );
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document.write( "You should find that<BR>\n" );
document.write( "P(-1.6421 < z < 4.1052) = 0.9497<BR>\n" );
document.write( "which leads back to<BR>\n" );
document.write( "P(0.22 < phat < 0.43) = <font color=red size=4>0.9497</font><BR>\n" );
document.write( "This value is approximate.\r<BR>\n" );
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document.write( "If we randomly picked phat values from the phat distribution, then there's roughly a 94.97% chance of getting a phat in the interval 0.22 < phat < 0.43<BR>\n" );
document.write( "phat = sample proportion<BR>\n" );
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