document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1179208>Question 1179208</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Hi! Please help with the following question! Thank you so much!\r<BR>\n" );
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document.write( " A coffee machine dispenses normally distributed amounts of coffee with a mean of 12 ounces and a standard deviation of 0.2 ounce. If a sample of 9 cups is selected, find the probability that the mean of the sample will be greater than 12.1 ounces. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #850259 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=850259\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Here's how to solve this problem:\r<BR>\n" );
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document.write( "**1. Understand the Problem:**\r<BR>\n" );
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document.write( "* We're dealing with a normally distributed population (coffee dispensed) with a known mean and standard deviation.<BR>\n" );
document.write( "* We're interested in the probability of a sample mean being greater than a certain value.<BR>\n" );
document.write( "* This requires using the Central Limit Theorem and z-scores.\r<BR>\n" );
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document.write( "**2. Define the Parameters:**\r<BR>\n" );
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document.write( "* Population mean (μ): 12 ounces<BR>\n" );
document.write( "* Population standard deviation (σ): 0.2 ounces<BR>\n" );
document.write( "* Sample size (n): 9<BR>\n" );
document.write( "* Sample mean (x̄): 12.1 ounces\r<BR>\n" );
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document.write( "**3. Calculate the Standard Error of the Mean:**\r<BR>\n" );
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document.write( "* The standard error (SE) is the standard deviation of the sampling distribution of the mean.<BR>\n" );
document.write( "* SE = σ / √n = 0.2 / √9 = 0.2 / 3 ≈ 0.0667 ounces\r<BR>\n" );
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document.write( "**4. Calculate the Z-Score:**\r<BR>\n" );
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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) / SE = (12.1 - 12) / 0.0667 = 0.1 / 0.0667 ≈ 1.5\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 P(x̄ > 12.1), which is the same as P(z > 1.5).<BR>\n" );
document.write( "* Using a z-table or calculator:<BR>\n" );
document.write( "    * P(z < 1.5) ≈ 0.9332<BR>\n" );
document.write( "    * P(z > 1.5) = 1 - P(z < 1.5) = 1 - 0.9332 ≈ 0.0668\r<BR>\n" );
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document.write( "**Answer:**\r<BR>\n" );
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document.write( "The probability that the mean of the sample will be greater than 12.1 ounces is approximately 0.0668.<BR>\n" );
document.write( " </SPAN>\n" );
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