document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1202886>Question 1202886</A>: <I><SPAN STYLE=\"word-break: break-all;\"> #8-13\r<BR>\n" );
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document.write( "The mean amount purchased by a typical customer at Churchill's Grocery Store is $23.50 with a standard deviation of $6.00. Assume the distribution of amounts purchased follows the normal distribution. For a sample of 52 customers, answer the following questions.\r<BR>\n" );
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document.write( "a. What is the likelihood the sample mean is at least $25.50? (Round your z-value to 2 decimal places and final answer to 4 decimal places.)<BR>\n" );
document.write( "b. What is the likelihood the sample mean is greater than $22.50 but less than $25.50? (Round your z-value to 2 decimal places and final answer to 4 decimal places.)<BR>\n" );
document.write( "c. Within what limits will 95 percent of the sample means occur? (Round your answers to 2 decimal places.) ____ and _____.<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 #838088 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=838088\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> m = mean = 23.5<BR>\n" );
document.write( "d = standard devition = 6<BR>\n" );
document.write( "n = 52 = sample size<BR>\n" );
document.write( "s = standard error = d / sqrt(n) = .83205\r<BR>\n" );
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document.write( "a. What is the likelihood the sample mean is at least $25.50? (Round your z-value to 2 decimal places and final answer to 4 decimal places.)\r<BR>\n" );
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document.write( "z = (x-m)/s = (25.5 - 23.5) / .83205 = 2.4<BR>\n" );
document.write( "area to the left of that = .9918<BR>\n" );
document.write( "area to the right of that = 1 - .9918 = .0082.<BR>\n" );
document.write( "the probability of getting a sample mean greater than 25.5 = .0082.\r<BR>\n" );
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document.write( "b. What is the likelihood the sample mean is greater than $22.50 but less than $25.50? (Round your z-value to 2 decimal places and final answer to 4 decimal places.)\r<BR>\n" );
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document.write( "z-score for lower bound = (22.5 - 23.5) / .83205 = -1.2<BR>\n" );
document.write( "area to the left of that z-score = .1151<BR>\n" );
document.write( "z-score for upper gound = (25.5 - 23.5) / .83205 = 2.4<BR>\n" );
document.write( "area to the left of that z-score = .9918<BR>\n" );
document.write( "area in between = .99918 minus .1151 = .8767<BR>\n" );
document.write( "the probability of getting a sample mean between 22.5 and 25.5 is equal to .8767.\r<BR>\n" );
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document.write( "c. Within what limits will 95 percent of the sample means occur? (Round your answers to 2 decimal places.)\r<BR>\n" );
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document.write( "z-score formula for lower bound is -1.96 = (x - 23.5) / .83205.<BR>\n" );
document.write( "solve for x to get x = -1.96 * .83205 + 23.5 = 21.87.<BR>\n" );
document.write( "z-score formula for upper bound is 1.96 = (x - 23.5) / .83205.<BR>\n" );
document.write( "solve for x to get x = 1.96 * .83205 + 23.5 = 25.13<BR>\n" );
document.write( "95% confidence interval is from 21.87 to 25.13\r<BR>\n" );
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document.write( "here's what the solutions look like on the z-score graphing calculator.\r<BR>\n" );
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document.write( "when using z-score, the mean is 0 and the standard deviation is 1.<BR>\n" );
document.write( "when using raw scores, the mean is 23.5 and the standard deviation is .83205.\r<BR>\n" );
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document.write( "note that what is titled standard deviation is the standare error.<BR>\n" );
document.write( "the standard error is another name for the standard deviation of the distribution of sample means.\r<BR>\n" );
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document.write( "solution to problem a.\r<BR>\n" );
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document.write( "solution to problem b.\r<BR>\n" );
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document.write( "solution to problem c.\r<BR>\n" );
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document.write( "any difference in the solution from z-score and the solution from raw scores is due to rounding only.\r<BR>\n" );
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