document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1204333>Question 1204333</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A population of values has a normal distribution with = 131.2 and o = 20.3. You intend to draw a random<BR>\n" );
document.write( "sample of size n = 18.<BR>\n" );
document.write( "Find the probability that a single randomly selected value is less than 142.7.<BR>\n" );
document.write( "P(X < 142.7) =<BR>\n" );
document.write( "Find the probability that a sample of size n = 18 is randomly selected with a mean less than 142.7.<BR>\n" );
document.write( "P(M < 142.7) =<BR>\n" );
document.write( "Enter your answers as numbers accurate to 4 decimal places. Answers obtained using exact z-scores or z-scores rounded to 3 decimal places are accepted. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #840517 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=840517\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> population meaan is 131.2<BR>\n" );
document.write( "population standard deviation is 20.3 (i believe that you meant standard deviation).<BR>\n" );
document.write( "sample size is 18.\r<BR>\n" );
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document.write( "z-score formula is:<BR>\n" );
document.write( "z = (x-m)/s<BR>\n" );
document.write( "z is the z-score<BR>\n" );
document.write( "x is the raw score<BR>\n" );
document.write( "m is the raw mean<BR>\n" );
document.write( "s is the standard deviation if you are looking for an individual score.<BR>\n" );
document.write( "s is the standard error if you are looking for the mean of a sample of given size.\r<BR>\n" );
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document.write( "quesion 1.<BR>\n" );
document.write( "Find the probability that a single randomly selected value is less than 142.7.\r<BR>\n" );
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document.write( "you use the z-score formula with the standard deviation to find the probability of finding a single score less than 142.7\r<BR>\n" );
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document.write( "z = (x-m) / s becomes z = (142.7 - 131.2) / 20.3.<BR>\n" );
document.write( "you get z = .566502.<BR>\n" );
document.write( "the probability of getting a z-score less than that is equal to .714474.<BR>\n" );
document.write( "round to 4 decimal places to get .7145.\r<BR>\n" );
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document.write( "question 2.<BR>\n" );
document.write( "Find the probability that a sample of size n = 18 is randomly selected with a mean less than 142.7.\r<BR>\n" );
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document.write( "use the z-score formula with the standard error to find the probability that the mean of a sample of size 18 is less than 142.7.\r<BR>\n" );
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document.write( "standard error = standard deviation / sqrt(sample size) = 20.3 / sqrt(18) = 4.784756.\r<BR>\n" );
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document.write( "z = (x - m) / s becomes:<BR>\n" );
document.write( "z = (142.7 - 131.2) / 4.784756.<BR>\n" );
document.write( "you get z = 2.403466<BR>\n" );
document.write( "the probability of getting a z-score less than that is equal to .991880.<BR>\n" );
document.write( "round to 4 decimal places to get .9919.\r<BR>\n" );
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document.write( "your solutions are:\r<BR>\n" );
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document.write( "the probability that a single randomly selected value is less than 142.7 is equal to .7145.\r<BR>\n" );
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document.write( "the probability that a sample of size n = 18 is randomly selected with a mean less than 142.7 is equal to .9919.\r<BR>\n" );
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document.write( "note that the standard error is defined as the standard deviation of the distribution of sample means.<BR>\n" );
document.write( "that is not the same as the standard deviation of individual elements.<BR>\n" );
document.write( "as the samaple size gets larger, the standard error becomes smaller.<BR>\n" );
document.write( "that's bgecause the sample mean is closer to the population mean when the sample size is larger.\r<BR>\n" );
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document.write( "it can be dificult to visualize, but what you need to remember is:\r<BR>\n" );
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document.write( "if you are looking for the standard deviation of a single element in a set of data, then use the standard deviation.\r<BR>\n" );
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document.write( "if you are looking for the standard deviation of the mean of a sample of specified size from that set of data, then use the standard error.\r<BR>\n" );
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document.write( "note that, if the sample size is 1, then the standard error becomes the standard deviation because standard error = standard deviation / sqrt(1) = standard deviation / 1 = standard deviation.\r<BR>\n" );
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document.write( "i used a z-score calculator online to show you the difference between the z-sore of an individual element and the z-score of a sample mean of a sample of size 18.\r<BR>\n" );
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document.write( "the firest graph uses the standard deviation.<BR>\n" );
document.write( "the second graph uses the standrd error.\r<BR>\n" );
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