document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=648028>Question 648028</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A normal population has a mean of 75 and a standard deviation of 5. You select a sample of 40. Compute the probability the sample mean is: <BR>\n" );
document.write( "a. Less than 74.<BR>\n" );
document.write( "b. Between 74 and 76.<BR>\n" );
document.write( "c. Between 76 and 77.<BR>\n" );
document.write( "d. Greater than 77.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "I would love to get the answer so I can print it out and use it as a guide. \r<BR>\n" );
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document.write( "Thank you so much! </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #406533 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=406533\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> population mean = 75<BR>\n" );
document.write( "population standard deviation = 5<BR>\n" );
document.write( "sample size = 40<BR>\n" );
document.write( "i'll round all intermediate results to 3 decimal places and found the final answer to 2 decimal places.<BR>\n" );
document.write( "z-score = actual score minus population score divided by standard error.<BR>\n" );
document.write( "standard error = population standard deviation divided by square root of sample size.<BR>\n" );
document.write( "for this problem, standard error = 5/sqrt(40) = .791<BR>\n" );
document.write( "first you need to find the z-score and then you need to find the area under the normal distribution curve that applies.<BR>\n" );
document.write( "use of a z-score table is mandatory.<BR>\n" );
document.write( "there are different types, but the type show in this link is usually the best type to use because it's fairly easy to figure out what you need.<BR>\n" );
document.write( "<a  rel=nofollow HREF= \"http://lilt.ilstu.edu/dasacke/eco148/ztable.htm\" target = \"_blank\">http://lilt.ilstu.edu/dasacke/eco148/ztable.htm</a><BR>\n" );
document.write( "you need to compute the probability that the score is:<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "a. Less than 74.<BR>\n" );
document.write( "z-score = (74-75)/.791 = -1/.791 = -1.264 which can be rounded to -1.26 since the accuracy of the z-score tables is only 2 decimal digits.<BR>\n" );
document.write( "you look into the z-score to find the area under the distribution curve that is to the left of a z-score of -1.26.<BR>\n" );
document.write( "you look down the first column until you find -1.2 and then you look for the entry in the 7th column on that row.<BR>\n" );
document.write( "the entry you will find is .1038.<BR>\n" );
document.write( "that the area under the normal distribution curve to the left of a z-score of -1.26.<BR>\n" );
document.write( "this means that the probability of getting a z-score less than or equal to -1.26 is equal to .1038 or 10.38%.<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "b. Between 74 and 76.<BR>\n" );
document.write( "you need to find 2 z-scores.<BR>\n" );
document.write( "you need to find a z-score for an actual score of 74 and a z-score for an actual score of 76.<BR>\n" );
document.write( "z-score for 74 is:<BR>\n" );
document.write( "(74-75)/.791 = -1/.791 = -1.26 (rounded to 2 decimal digits).<BR>\n" );
document.write( "z-score for 76 is:<BR>\n" );
document.write( "(76-75)/.791 = 1/.791 = 1.26 (rounded to 2 decimal digits).<BR>\n" );
document.write( "you want the area under the normal distribution curve that is between a z-score of -1.26 and a z-score of 1.26<BR>\n" );
document.write( "that would be the area under the normal distribution curve to the left of a z-score of 1.26 minus the area under the normal distribution curve to the left of a z-score of -1.26.<BR>\n" );
document.write( "area to the left of a z-score of 1.26 is equal to .8962<BR>\n" );
document.write( "area to the left of a z-score of -1.26 is equal to .1038<BR>\n" );
document.write( ".8962 minus .1038 equals .7924<BR>\n" );
document.write( "that's the area between a z-score of -1.26 and 1.26 which is equal to the probability that you will get a z-score between -1.26 and +1.26 which is equal to the probability that you will get an actual score between 74 and 76 if the population mean is 75 and the population standard deviation is 5 and your sample size is 40.<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "c. Between 76 and 77.<BR>\n" );
document.write( "this is done in the same manner as between 74 and 76.<BR>\n" );
document.write( "you get the z-score for 76 and a z-score of 77.<BR>\n" );
document.write( "you then look up the area in the z-score table for a z-score of 76 and a z-score of 77.<BR>\n" );
document.write( "you then subtract the area to the left of a z-score of 76 from the area to the left of a z-score of 77 to get the difference which is the area between them which is the probability of getting a z-score between them.<BR>\n" );
document.write( "the numbers come out as follows:<BR>\n" );
document.write( "z-score of 76 = (76-75)/.791 = 1/.791 = 1.26<BR>\n" );
document.write( "z-score of 77 = (77-75)/.791 = 2/.791 = 2.53<BR>\n" );
document.write( "area to the left of a z-score of 1.26 = .8962<BR>\n" );
document.write( "area to the left of a z-score of 2.53 = .9943<BR>\n" );
document.write( "area between a z-score of 76 and 77 = .9943 - .8962 = .0981<BR>\n" );
document.write( "probability of getting a z-score between 1.26 and 2.53 = .0981<BR>\n" );
document.write( "that's the probability of getting an actual score between 76 and 77 if the population mean is 75 and the population standard deviation is 5 and the sample size is 40.<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "d. Greater than 77. <BR>\n" );
document.write( "here you want to find the area to the left of a z-score for 77 and then take 1 minus that to get the area to the right of that z-score.<BR>\n" );
document.write( "the z-score is (77-75)/791 = 2.53<BR>\n" );
document.write( "the area to the left of that z-score is .9943<BR>\n" );
document.write( "1 - .9943 = .0057<BR>\n" );
document.write( "that's the probability of getting a z-score greater than 2.53 which is the probability of getting a raw score greater than 77 if the population mean is 75 and the population standard deviation is 5 and the sample size is 40.<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "you are dealing with a sample mean and a distribution of sample means.<BR>\n" );
document.write( "the raw scores above are the sample means.<BR>\n" );
document.write( "the popuation mean is 75<BR>\n" );
document.write( "the population standard deviation is 5<BR>\n" );
document.write( "the sample size is 40.<BR>\n" );
document.write( "the standard error is equal to 5/sqrt(40) which is equal to .791<BR>\n" );
document.write( "the standard error is the standard deviation of the distribution of sample means.<BR>\n" );
document.write( "it is affected by the sample size.<BR>\n" );
document.write( "the larger the sample size, the smaller the standard error.<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "remember - difference s-score tables work different ways.<BR>\n" );
document.write( "the one i showed you is one of the easiest to follow.<BR>\n" );
document.write( "always check the table you are using to find out what they are assuming because they don't all work the same way.<BR>\n" );
document.write( " </SPAN>\n" );
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