document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1174328>Question 1174328</A>: <I><SPAN STYLE=\"word-break: break-all;\"> In  a  survey  of  a  company,  mean  salary  of  employees  is  MYR29,321  with  standard deviation  of  MYR2,120.  Consider  the  sample  of  100  employees  and  find  the probability  their  mean  salary  will  be  less  than  MYR29,000?  (Hint:  use  central  limit theorem).  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #850697 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=850697\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Here's how to solve this problem using the Central Limit Theorem:\r<BR>\n" );
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document.write( "**1. Understand the Central Limit Theorem**\r<BR>\n" );
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document.write( "* The Central Limit Theorem states that the distribution of sample means approaches a normal distribution as the sample size increases, regardless of the population's distribution.<BR>\n" );
document.write( "* The mean of the sample means (μ_x̄) is equal to the population mean (μ).<BR>\n" );
document.write( "* The standard deviation of the sample means (standard error, σ_x̄) is equal to the population standard deviation (σ) divided by the square root of the sample size (n).\r<BR>\n" );
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document.write( "**2. Given Information**\r<BR>\n" );
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document.write( "* Population mean (μ) = MYR 29,321<BR>\n" );
document.write( "* Population standard deviation (σ) = MYR 2,120<BR>\n" );
document.write( "* Sample size (n) = 100<BR>\n" );
document.write( "* Sample mean (x̄) = MYR 29,000\r<BR>\n" );
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document.write( "**3. Calculate the Standard Error**\r<BR>\n" );
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document.write( "* Standard error (σ_x̄) = σ / √n = 2120 / √100 = 2120 / 10 = 212\r<BR>\n" );
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document.write( "**4. Calculate the Z-score**\r<BR>\n" );
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document.write( "* Z = (x̄ - μ) / σ_x̄ = (29000 - 29321) / 212 = -321 / 212 ≈ -1.514\r<BR>\n" );
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document.write( "**5. Find the Probability**\r<BR>\n" );
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document.write( "* We want to find the probability that the sample mean is less than MYR 29,000, i.e., P(x̄ < 29000) or P(Z < -1.514).<BR>\n" );
document.write( "* Using a standard normal distribution table or a calculator (or the provided python code), we find that P(Z < -1.514) ≈ 0.06499.\r<BR>\n" );
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document.write( "**Answer:**\r<BR>\n" );
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document.write( "The probability that the mean salary of a sample of 100 employees will be less than MYR 29,000 is approximately 0.0650 or 6.5%.<BR>\n" );
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