document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1183363>Question 1183363</A>: <I><SPAN STYLE=\"word-break: break-all;\"> For a certain species of butterfly, the length of a wing (at its longest point) is normally distributed with a mean of 1 cm and a standard deviation of 0.2 cm.\r<BR>\n" );
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document.write( "Which is MORE LIKELY: a.) a single butterfly has wings that are shorter than 0.8 cm, or b.) a sample of 50 butterflies has an average wing length that is shorter than 0.8 cm.\r<BR>\n" );
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document.write( "Explain your reasoning.  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #813647 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=813647\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> with a sample of one butterfly, you use the standard deviation.\r<BR>\n" );
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document.write( "the population mean is 1 cm with a standard deviation of .2 cm.\r<BR>\n" );
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document.write( "the single butterfly has wings that are shorter than .8 cm.\r<BR>\n" );
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document.write( "the z-score formula 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<BR>\n" );
document.write( "m is the mean<BR>\n" );
document.write( "s is the standard deviation when you are dealing with a sample of one.<BR>\n" );
document.write( "s is the standard error when you are dealing with the mean of a sample of size greater than 1.\r<BR>\n" );
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document.write( "for the sample of 1, the formula becomes z = (.8 - 1) / .2 = -1.\r<BR>\n" );
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document.write( "the probability of getting a z-score less than -1 would be equal to .1586552596.\r<BR>\n" );
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document.write( "with a sample of 50 butterflies that has an average wing length that is shorter than .8 cm, the z-score becomes:\r<BR>\n" );
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document.write( "z = (.8 - 1) / s\r<BR>\n" );
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document.write( "s is the standard error.<BR>\n" );
document.write( "the standard error is equal to the standard deviation divided by  the square root of the sample size.<BR>\n" );
document.write( "s = .2 / sqrt(50) = .02828\r<BR>\n" );
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document.write( "formula becomes z = (.8 - 1) / .02828 = -7.072135785.\r<BR>\n" );
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document.write( "the mean of the sample being less than .8 is decidedly LESS likely than the length of the wings of one butterfly being less than .8.\r<BR>\n" );
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document.write( "the mean of the sample of 50 contains measurements of 50 butterflies, rather than just one.\r<BR>\n" );
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document.write( "the standard error, which is the standard deviation of the distribution of sample means, will become smaller when the sample size is larger.\r<BR>\n" );
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document.write( "that's why the standard error formula is equal to the standard deviation divided by the square root of the sample size.\r<BR>\n" );
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document.write( "with the standard error being less than the standard deviation, the same value of x will have a higher z-score which means the probability of getting a score less than that will be less.\r<BR>\n" );
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document.write( "-.2 / .02828 give you a higher z-score than -.2 / .2<BR>\n" );
document.write( "a higher z-score means less probability of getting a score less than .8.<BR>\n" );
document.write( "the one with the lower z-score is more probable.<BR>\n" );
document.write( "that's the measurement from the single element rather than the mean of the sample of 50 elements.\r<BR>\n" );
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document.write( "you will see more variability in the length of the wings from a sample of one butterfly than from the mean of a sample of 50 butterflies.\r<BR>\n" );
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document.write( "because of the greater variability, the probability that you will find a butterfly with wings less than .8 cm is greater than what you will find with the mean of a sample of 50 butterflies.\r<BR>\n" );
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document.write( " </SPAN>\n" );
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