document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1170244>Question 1170244</A>: <I><SPAN STYLE=\"word-break: break-all;\"> In a certain stats class, the marks obtained by students on a class test followed a normal distribution with a mean of 68% and a standard deviation of 10%. What is the probability that the mean test mark from a sample of 25 students from the class was more than 72%?<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 #851233 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=851233\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let's break down this problem step by step.\r<BR>\n" );
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document.write( "**Understanding the Problem**\r<BR>\n" );
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document.write( "We are dealing with the sampling distribution of the mean. We're given:\r<BR>\n" );
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document.write( "* Population mean (μ) = 68%<BR>\n" );
document.write( "* Population standard deviation (σ) = 10%<BR>\n" );
document.write( "* Sample size (n) = 25<BR>\n" );
document.write( "* We want to find the probability that the sample mean (x̄) > 72%\r<BR>\n" );
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document.write( "**Key Concepts**\r<BR>\n" );
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document.write( "* **Sampling Distribution of the Mean:** When we take multiple samples from a population, the means of those samples form a distribution.<BR>\n" );
document.write( "* **Central Limit Theorem:** If the population is normally distributed or if the sample size is large enough (n ≥ 30), the sampling distribution of the mean is approximately normal.<BR>\n" );
document.write( "* **Standard Error of the Mean (SEM):** The standard deviation of the sampling distribution of the mean is given by σ / √n.\r<BR>\n" );
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document.write( "**Calculations**\r<BR>\n" );
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document.write( "1.  **Calculate the Standard Error of the Mean (SEM):**<BR>\n" );
document.write( "    SEM = σ / √n = 10% / √25 = 10% / 5 = 2%\r<BR>\n" );
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document.write( "2.  **Calculate the Z-score:**<BR>\n" );
document.write( "    The z-score tells us how many standard errors the sample mean is away from the population mean.<BR>\n" );
document.write( "    z = (x̄ - μ) / SEM = (72% - 68%) / 2% = 4% / 2% = 2\r<BR>\n" );
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document.write( "3.  **Find the Probability:**<BR>\n" );
document.write( "    We want to find P(x̄ > 72%), which is equivalent to finding P(Z > 2).<BR>\n" );
document.write( "    Using a standard normal distribution table or a calculator, we find that:<BR>\n" );
document.write( "    P(Z < 2) ≈ 0.9772<BR>\n" );
document.write( "    Therefore, P(Z > 2) = 1 - P(Z < 2) = 1 - 0.9772 = 0.0228\r<BR>\n" );
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document.write( "**Answer**\r<BR>\n" );
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document.write( "The probability that the mean test mark from a sample of 25 students was more than 72% is approximately 0.0228.<BR>\n" );
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