document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=964030>Question 964030</A>: <I><SPAN STYLE=\"word-break: break-all;\"> a dice is thrown 9000 times and 3 or 4 is observed 3240 times can we say dice is fair. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #588948 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=588948\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> probably not, based on the following test.\r<BR>\n" );
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document.write( "n = 9000<BR>\n" );
document.write( "p (3 or 4) = 1/6 + 1/6 = 2/6 = 1/3<BR>\n" );
document.write( "(1-p) = 2/3\r<BR>\n" );
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document.write( "mean = n * p = 9000 * 1/3 = 3000.\r<BR>\n" );
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document.write( "standard error equal sqrt (n * p * (1-p) = sqrt ( 9000 * 1/3 * 2/3) = sqrt(2000) = somewhere between 44.72 and 44.73, the exact value not being important for this problem.\r<BR>\n" );
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document.write( "a raw score of 3240 will generate a z-score of (x-m) / se which would be equal to:\r<BR>\n" );
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document.write( "z = (3240 - 3000) / 44.72 or 44.73.\r<BR>\n" );
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document.write( "therefore z = 5.367 or 5.366.\r<BR>\n" );
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document.write( "both are more than 3 standard deviations from the mean.\r<BR>\n" );
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document.write( "the probability of getting a z-score greater than 3 standard deviations from the mean would be equal to .00135.\r<BR>\n" );
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document.write( "this is less than .05 and also less than .01 which are the probabilities normally used as criteria.\r<BR>\n" );
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document.write( "the .05 is normally used at the .95 confidence level for a one tailed test.\r<BR>\n" );
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document.write( "the .01 is normally used as the .99 confidence level for a one tailed test.\r<BR>\n" );
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document.write( "bottom line is the probability of getting such a score out of 9000 tosses just based on random differences between different samples is very remote and therefore not considered feasible.\r<BR>\n" );
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document.write( "this indicates that the dice are probably biased in some way.\r<BR>\n" );
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document.write( " </SPAN>\n" );
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