document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1174398>Question 1174398</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The mean salary offered to students who are graduating from Coastal State University this year is $24,260, with a standard deviation of $3712. A random sample of 75 Coastal State students graduating this year has been selected. What is the probability that the mean salary offer for these 75 students is $24,000 or less?<BR>\n" );
document.write( "Carry your intermediate computations to at least four decimal places. Round your answer to at least three decimal places.\r<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 #850670 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1959)</B></A><img src=\"/images/stars/stars-12.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=850670\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> **1. Calculate the Standard Error:**\r<BR>\n" );
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document.write( "* The standard error of the mean (SEM) is the standard deviation of the sampling distribution of the mean.<BR>\n" );
document.write( "* SEM = σ / √n <BR>\n" );
document.write( "   where:<BR>\n" );
document.write( "     * σ = population standard deviation ($3712)<BR>\n" );
document.write( "     * n = sample size (75)<BR>\n" );
document.write( "* SEM = 3712 / √75 ≈ 428.3193\r<BR>\n" );
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document.write( "**2. Calculate the z-score:**\r<BR>\n" );
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document.write( "* The z-score measures how many standard errors the sample mean is away from the population mean.<BR>\n" );
document.write( "* z = (x̄ - μ) / SEM<BR>\n" );
document.write( "   where:<BR>\n" );
document.write( "     * x̄ = sample mean ($24,000)<BR>\n" );
document.write( "     * μ = population mean ($24,260)<BR>\n" );
document.write( "     * SEM = standard error of the mean (428.3193)<BR>\n" );
document.write( "* z = (24000 - 24260) / 428.3193 ≈ -0.6070\r<BR>\n" );
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document.write( "**3. Find the Probability:**\r<BR>\n" );
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document.write( "* We want to find the probability that the sample mean is $24,000 or less, which is equivalent to finding the area to the left of the z-score of -0.6070 in the standard normal distribution.<BR>\n" );
document.write( "* Using a z-table or calculator, we find that the probability is approximately 0.272.\r<BR>\n" );
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document.write( "**Therefore, the probability that the mean salary offer for these 75 students is $24,000 or less is approximately 0.272.** </SPAN>\n" );
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