document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1170505>Question 1170505</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A machine fills pop bottles with a standard deviation of 25 mL no matter what the setting is for the mean. At how many mililiters should the mean be set, so that 90% of the bottles will contain at least 500 mL? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #851167 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=851167\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let X be the amount of liquid filled in the pop bottles. We are given:\r<BR>\n" );
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document.write( "* Standard deviation (σ) = 25 mL<BR>\n" );
document.write( "* We want 90% of the bottles to contain at least 500 mL, which means P(X ≥ 500) = 0.90.\r<BR>\n" );
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document.write( "We want to find the mean (μ) such that this condition is satisfied.\r<BR>\n" );
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document.write( "**1. Convert to Standard Normal Distribution**\r<BR>\n" );
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document.write( "We can standardize the random variable X using the formula:\r<BR>\n" );
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document.write( "Z = (X - μ) / σ\r<BR>\n" );
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document.write( "where Z follows a standard normal distribution (mean 0, standard deviation 1).\r<BR>\n" );
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document.write( "We want P(X ≥ 500) = 0.90. Let's rewrite this in terms of Z:\r<BR>\n" );
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document.write( "P((X - μ) / σ ≥ (500 - μ) / 25) = 0.90<BR>\n" );
document.write( "P(Z ≥ (500 - μ) / 25) = 0.90\r<BR>\n" );
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document.write( "**2. Find the Z-score**\r<BR>\n" );
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document.write( "We need to find the z-score such that the area to the right of it is 0.90. This means the area to the left of it is 1 - 0.90 = 0.10.\r<BR>\n" );
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document.write( "Using a standard normal distribution table or a calculator, we find the z-score corresponding to a cumulative probability of 0.10. This z-score is approximately -1.28.\r<BR>\n" );
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document.write( "So, (500 - μ) / 25 = -1.28.\r<BR>\n" );
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document.write( "**3. Solve for μ**\r<BR>\n" );
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document.write( "Now, we solve for μ:\r<BR>\n" );
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document.write( "500 - μ = -1.28 * 25<BR>\n" );
document.write( "500 - μ = -32<BR>\n" );
document.write( "μ = 500 + 32<BR>\n" );
document.write( "μ = 532\r<BR>\n" );
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document.write( "**4. Answer**\r<BR>\n" );
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document.write( "The mean should be set to 532 mL.\r<BR>\n" );
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document.write( "**Therefore, the mean should be set to 532 mL so that 90% of the bottles contain at least 500 mL.**<BR>\n" );
document.write( " </SPAN>\n" );
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