document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1195144>Question 1195144</A>: <I><SPAN STYLE=\"word-break: break-all;\"> 3.	(a) The starting salary among 1999 graduates of the Evangelican Business School has a standard deviation of $17,000. If you randomly survey 40 students and average their starting salaries, what is the probability that the average among these students will greaterthan the average among all students?<BR>\n" );
document.write( "(b)	What is the probability that the average in the sample will exceed the average amo all students by more than $5,000?<BR>\n" );
document.write( "(c)	What is the probability that the average in the sample will differ from the avera among all students by more than $5,000?\r<BR>\n" );
document.write( "\n" );
document.write( " </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #848436 by </B><A HREF=/tutors/aboutme.mpl?userid=ElectricPavlov><B>ElectricPavlov(122)</B></A><img src=\"/images/stars/stars-09.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=ElectricPavlov><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=848436\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> **a) Probability that the average among these students will be greater than the average among all students**\r<BR>\n" );
document.write( "\n" );
document.write( "* **Assumptions:**<BR>\n" );
document.write( "    * We assume that the starting salaries of all 1999 graduates are normally distributed.<BR>\n" );
document.write( "    * We are comparing the sample mean to the population mean.\r<BR>\n" );
document.write( "\n" );
document.write( "* **Central Limit Theorem:** <BR>\n" );
document.write( "    * The sampling distribution of the sample mean will be approximately normally distributed with:<BR>\n" );
document.write( "        * Mean (μ_x̄) = μ (population mean)<BR>\n" );
document.write( "        * Standard Deviation (σ_x̄) = σ / √n <BR>\n" );
document.write( "            * where:<BR>\n" );
document.write( "                * σ = population standard deviation ($17,000)<BR>\n" );
document.write( "                * n = sample size (40)<BR>\n" );
document.write( "            * σ_x̄ = $17,000 / √40 ≈ $2690.87\r<BR>\n" );
document.write( "\n" );
document.write( "* **Probability of Sample Mean Exceeding Population Mean:**<BR>\n" );
document.write( "    * Since we're comparing the sample mean to the population mean, we are essentially asking for the probability that the sample mean is greater than the population mean. <BR>\n" );
document.write( "    * In a normal distribution, 50% of the data lies above the mean. <BR>\n" );
document.write( "    * Therefore, the probability that the average among the 40 students will be greater than the average among all students is **50%**.\r<BR>\n" );
document.write( "\n" );
document.write( "**b) Probability that the average in the sample will exceed the average among all students by more than $5,000**\r<BR>\n" );
document.write( "\n" );
document.write( "* **Calculate the z-score:**<BR>\n" );
document.write( "    * z = (X - μ_x̄) / σ_x̄<BR>\n" );
document.write( "        * where:<BR>\n" );
document.write( "            * X = Difference in means ($5,000)<BR>\n" );
document.write( "            * μ_x̄ = 0 (since we're comparing the sample mean to the population mean)<BR>\n" );
document.write( "            * σ_x̄ = $2690.87\r<BR>\n" );
document.write( "\n" );
document.write( "    * z = ($5,000 - $0) / $2690.87 <BR>\n" );
document.write( "    * z ≈ 1.86\r<BR>\n" );
document.write( "\n" );
document.write( "* **Find the Probability:**<BR>\n" );
document.write( "    * Use a standard normal distribution table (z-table) to find the probability that z is greater than 1.86. <BR>\n" );
document.write( "    * P(z > 1.86) ≈ 0.0314\r<BR>\n" );
document.write( "\n" );
document.write( "* **Therefore, the probability that the average in the sample will exceed the average among all students by more than $5,000 is approximately 3.14%.**\r<BR>\n" );
document.write( "\n" );
document.write( "**c) Probability that the average in the sample will differ from the average among all students by more than $5,000**\r<BR>\n" );
document.write( "\n" );
document.write( "* This includes both cases: <BR>\n" );
document.write( "    * Sample mean is greater than population mean by more than $5,000 (calculated in part b)<BR>\n" );
document.write( "    * Sample mean is less than population mean by more than $5,000\r<BR>\n" );
document.write( "\n" );
document.write( "* **Due to the symmetry of the normal distribution:**<BR>\n" );
document.write( "    * Probability of differing by more than $5,000 = 2 * P(z > 1.86) <BR>\n" );
document.write( "    * Probability = 2 * 0.0314 = 0.0628\r<BR>\n" );
document.write( "\n" );
document.write( "* **Therefore, the probability that the average in the sample will differ from the average among all students by more than $5,000 is approximately 6.28%.**\r<BR>\n" );
document.write( "\n" );
document.write( "**Key Assumptions:**\r<BR>\n" );
document.write( "\n" );
document.write( "* The starting salaries of all 1999 graduates are normally distributed.<BR>\n" );
document.write( "* The sample of 40 students is a random sample from the population of all 1999 graduates.\r<BR>\n" );
document.write( "\n" );
document.write( "I hope this comprehensive explanation is helpful!<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );