document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1164883>Question 1164883</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Business Weekly conducted a survey of graduates from 30 top MBA programs. On the basis of the survey, assume the mean annual salary for graduates 10 years after graduation is $176,000. Assume the standard deviation is $32,000. Suppose you take a simple random sample of 48 graduates. Round all answers to four decimal places if necessary.\r<BR>\n" );
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document.write( "What is the distribution of  <BR>\n" );
document.write( "X?X~ N(____,_____)\r<BR>\n" );
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document.write( "What is the distribution of  <BR>\n" );
document.write( "¯x?¯x~ N(______,_______)\r<BR>\n" );
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document.write( "For a single randomly selected graduate, find the probability that her salary is between $178,172 and $184,181. \r<BR>\n" );
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document.write( "For a simple random sample of 48 graduates, find the probability that the average salary is between $178,172 and $184,181. \r<BR>\n" );
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document.write( "For part d), is the assumption of normal necessary? No or Yes </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #853507 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(2189)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=CPhill><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=CPhill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=853507\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> To solve this, we use the principles of the Normal Distribution and the **Central Limit Theorem**.\r<BR>\n" );
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document.write( "### a) What is the distribution of ?\r<BR>\n" );
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document.write( "The distribution of the individual annual salaries () follows the population mean and standard deviation.\r<BR>\n" );
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document.write( "### b) What is the distribution of ?\r<BR>\n" );
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document.write( "For a sample of size , the sampling distribution of the mean () is also normal, but with a smaller standard deviation (Standard Error).\r<BR>\n" );
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document.write( "* **Mean ():** <BR>\n" );
document.write( "* **Standard Error ():** \r<BR>\n" );
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document.write( "---\r<BR>\n" );
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document.write( "### c) For a single randomly selected graduate, find the probability that her salary is between $178,172 and $184,181.\r<BR>\n" );
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document.write( "We need to find . We calculate the -scores for both values:\r<BR>\n" );
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document.write( "* <BR>\n" );
document.write( "* \r<BR>\n" );
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document.write( "Using a standard normal distribution table or calculator:\r<BR>\n" );
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document.write( "**Probability = 0.0738**\r<BR>\n" );
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document.write( "---\r<BR>\n" );
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document.write( "### d) For a simple random sample of 48 graduates, find the probability that the average salary is between $178,172 and $184,181.\r<BR>\n" );
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document.write( "We use the Standard Error () to find the -scores for the sample mean:\r<BR>\n" );
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document.write( "* <BR>\n" );
document.write( "* \r<BR>\n" );
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document.write( "Using a standard normal distribution table or calculator:\r<BR>\n" );
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document.write( "**Probability = 0.2808**\r<BR>\n" );
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document.write( "---\r<BR>\n" );
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document.write( "### e) For part d), is the assumption of normal necessary?\r<BR>\n" );
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document.write( "**No**.<BR>\n" );
document.write( "Because the sample size () is greater than 30, the **Central Limit Theorem** ensures that the sampling distribution of the mean will be approximately normal even if the underlying population distribution is not.\r<BR>\n" );
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document.write( "Would you like me to explain how the probability would change if the sample size was smaller, say ? </SPAN>\n" );
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