document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1207237>Question 1207237</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A fish stall owner at a market place claims that the weight of a catfish he is selling is approximately normally distributed with a mean of 3 pounds and a standard deviation of 0.5 pounds. A random sample of 20 catfishes is selected.\r<BR>\n" );
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document.write( "What is the PROBABILITY that the average weight is at least 2.8 pounds?<BR>\n" );
document.write( "What is the PROBABILITY that the average weight is between 2.8 pounds and 3.2 pounds?	<BR>\n" );
document.write( "What is the AVERAGE WEIGHT of top 10% of catfish?<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #844991 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=844991\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> population mean is assumed to be 3 pounds.<BR>\n" );
document.write( "population standard deviation is assumed to be .5 pounds.\r<BR>\n" );
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document.write( "sample size is 20.\r<BR>\n" );
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document.write( "you have to use the standard error because you are looking for a distribution of sample means, rather than a distribution of sample elements.\r<BR>\n" );
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document.write( "standard error = standard deviation / sqrt(sample size) = .5 / sqrt(30) = .111803 rounded to 6 decimal places.\r<BR>\n" );
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document.write( "you are using the z-score because the standard deviation is taken from the population.\r<BR>\n" );
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document.write( "What is the PROBABILITY that the average weight is at least 2.8 pounds?\r<BR>\n" );
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document.write( "z = (2.8 - 3) / .111803 = -1.7889\r<BR>\n" );
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document.write( "area under the normal distribution curve to the right of that z-score is equal to .96318.\r<BR>\n" );
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document.write( "probability of getting a z-score greater than that is equal to .96318 rounded to 5 decimal places.\r<BR>\n" );
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document.write( "that's the probability of getting a sample with average weight greater than 2.8 pounds.\r<BR>\n" );
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document.write( "What is the PROBABILITY that the average weight is between 2.8 pounds and 3.2 pounds?\r<BR>\n" );
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document.write( "z-score for 2.8 pounds = (2.8 - 3) / .111803 = -1.7889.<BR>\n" );
document.write( "z-score for 3.2 pounds = (3.2 - 3) / .111803 = 1.7889.\r<BR>\n" );
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document.write( "area to the left of z-score 0f -1.7889 = .03682 rounded to 5 decimal places.<BR>\n" );
document.write( "area to the left of z-score of 1.7889 = .96318 rounded to 5 decimal places.<BR>\n" );
document.write( "area in between = larger area minus smaller area = .92636.\r<BR>\n" );
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document.write( "that's the probability of getting a z-score greater than -1.7889 and less then 1.7889.\r<BR>\n" );
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document.write( "that's the probability of getting a sample with average weight greater than 2.8 and less than 3.2 pounds.\r<BR>\n" );
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document.write( "What is the AVERAGE WEIGHT of top 10% of catfish?\r<BR>\n" );
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document.write( "look for a z-score that has 10% of the area under the normal distribution curve to the right of it.<BR>\n" );
document.write( "that's the same as a z-score with 90% of the area under the normal distribution curve to the left of it.\r<BR>\n" );
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document.write( "i get z-score = 1.28188.\r<BR>\n" );
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document.write( "use the z-score formula to find the raw score.\r<BR>\n" );
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document.write( "1.28188 = (x - 3) / .111803.<BR>\n" );
document.write( "solve for x to get x = 3.1433 rounded to 4 decimal places.\r<BR>\n" );
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document.write( "the probability of getting a sample with a mean that is in the top 10% of the possible sample means is getting a sample with a mean of 3.1433 rounded to 4 decimal places.\r<BR>\n" );
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document.write( "i used an online calculator to verify these answers.<BR>\n" );
document.write( "the calculator can be found at <a  rel=nofollow HREF= \"\" target = \"_blank\"></a>\r<BR>\n" );
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document.write( "i used the ti-04 plus calculator to get the answers.<BR>\n" );
document.write( "i used the calculator at <a  rel=nofollow HREF= \"\" target = \"_blank\"></a> to verify the answers were correct.<BR>\n" );
document.write( "the key was to calculate the standard error because you were looking for sample mean rather than a sample element.<BR>\n" );
document.write( "everything flowed from there.\r<BR>\n" );
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document.write( "here are the calculator results.\r<BR>\n" );
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