document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=844860>Question 844860</A>: <I><SPAN STYLE=\"word-break: break-all;\"> In a recent poll, 45% of survey respondents said that, if they only had one child, they would prefer the child to be a boy. Suppose you conducted a survey of 190 randomly selected students on your campus and find that 86 of them would prefer a boy.<BR>\n" );
document.write( "a) Use the normal approximation to the binomial to approximate the probability that, in a random sample of 190 students, at least 86 would prefer a boy, assuming the true percentage is 45%.<BR>\n" );
document.write( "b) Does this result contradict the poll? Explain  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #508998 by </B><A HREF=/tutors/aboutme.mpl?userid=Theo><B>Theo(13342)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Theo><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/cgi-bin/show-face.mpl?user=Theo><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=508998\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> normal approximation to binomial is:<BR>\n" );
document.write( "mean = n * p<BR>\n" );
document.write( "standard deviation = square root of (n * p * (1-p))<BR>\n" );
document.write( "n is equal to 190<BR>\n" );
document.write( "p is equal to .45<BR>\n" );
document.write( "n * p is equal to .45 * 190 = 85.5<BR>\n" );
document.write( "this is the probability that you would expect based on the knowledge that 45% of survey respondents said they would prefer a boy than a girl.<BR>\n" );
document.write( "in your survey you got 86 out of 190 and you want to know if this is statistically significant or if you can expect this deviation by chance.<BR>\n" );
document.write( "standard deviation is calculated as square root of (n * p * (n-p)) which becomes square root of (190 * .45 * .55) which is equal to 6.8575 rounded to 4 decimal places which is more than enough accuracy for what is required.<BR>\n" );
document.write( "without a continuity correction, your formula would be:<BR>\n" );
document.write( "z (86 > 85.5) =  (population mean - sample mean) / population standard deviation which becomes (86 - 85.5) / 6.8575 which is equal to .0729.<BR>\n" );
document.write( "this is so close to a z factor of 0 that there's no question that the results are well within limits and can very easily be due to chance.<BR>\n" );
document.write( "with a continuity correction, your formula would be:<BR>\n" );
document.write( "z (85.5 > 85.5) = 0<BR>\n" );
document.write( "a z factor of zero means that your sample mean is exactly the population mean.<BR>\n" );
document.write( "the probability that your score will be above 0 is 50%.<BR>\n" );
document.write( "since 5% is the cutoff, your well within the confidence interval limits.<BR>\n" );
document.write( "the following video might help you with the process as it demonstrates solving a similar type problem.<BR>\n" );
document.write( "<a  rel=nofollow HREF= \"http://www.youtube.com/watch?v=mdIUHzWydr8\" target = \"_blank\">http://www.youtube.com/watch?v=mdIUHzWydr8</a>\r<BR>\n" );
document.write( "<BR>\n" );
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document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );