document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1132378>Question 1132378</A>: <I><SPAN STYLE=\"word-break: break-all;\"> 1. Is It Unusual? A population is normally distributed with a mean of 100 and a standard deviation of 15. Determine if the following event is unusual. Explain your reasoning.<BR>\n" );
document.write( "a. mean of a sample of 3 is 115 or more <BR>\n" );
document.write( "b. mean of a sample of 20 is 105 or more </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #749437 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=749437\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> you need to calculate the standard error.<BR>\n" );
document.write( "the standard error is the standard deviation of the distribution of sample means.<BR>\n" );
document.write( "the larger the size of the sample, the smaller the standard error.\r<BR>\n" );
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document.write( "your population has a mean of 100 and a standard deviation of 15.\r<BR>\n" );
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document.write( "the standard error is equal to the standard deviation divided by the square root of the sample size.\r<BR>\n" );
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document.write( "for a sample size of 3, the standard error is equal to 15 / sqrt(3) = 8.66025...\r<BR>\n" );
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document.write( "for a sample size of 20, the standard error is equal to 15 / sqrt(20) = 3.35410...\r<BR>\n" );
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document.write( "you want to find the z-score and then determine the area to the right of that z-score.\r<BR>\n" );
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document.write( "the formula for z-wscore is z = (x - m) / s\r<BR>\n" );
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document.write( "z is the z-score<BR>\n" );
document.write( "x is the raw score you are comparing to the mean.<BR>\n" );
document.write( "m is the mean.<BR>\n" );
document.write( "s is the standard error.\r<BR>\n" );
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document.write( "for the sample of 3, you want to find out if a score of 115 is unusual.\r<BR>\n" );
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document.write( "the formula for z-score becomes z = (115 - 100) / 8.66025 = 1.73205...\r<BR>\n" );
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document.write( "the probability of getting a z-score greater than 1.73205 ..... is .041632....\r<BR>\n" );
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document.write( "what that says is that, if you took a very large number of samples of size 3, approximately 4.2% of them would have a mean greater than 115.\r<BR>\n" );
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document.write( "conversely, approximately 95.8% of them would have a mean less than 115.\r<BR>\n" );
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document.write( "with a sample size of 20 and a score of 105 or more, the z-score would be equal to (105 - 100) / 3.35410 = 1.490712...\r<BR>\n" );
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document.write( "a probability of getting a z-score greater than 1.490712... is .068018...\r<BR>\n" );
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document.write( "what that says is that, if you took a very large number of samples of size 20, approximately 6.8% of them would have a mean greater than 105.\r<BR>\n" );
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document.write( "conversely, approximately 93.2% of them would have a mean less than 105.\r<BR>\n" );
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document.write( "are either of these unusual?\r<BR>\n" );
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document.write( "it depends on what you consider unusual.\r<BR>\n" );
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document.write( "if you consider occurring 5% of the time or less is unusual, than the first one is unusual and the second one isn't.\r<BR>\n" );
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document.write( "if you consider occurring 10% of the time or less is unusual, than both are unusual.\r<BR>\n" );
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document.write( "if you consider occurring 2% of the time or less is unusual, than neither are unusual.\r<BR>\n" );
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document.write( " </SPAN>\n" );
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