document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1192784>Question 1192784</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A soda machine dispenses soda into 16-ounce cups. Tests shows that the actual amount of soda dispensed has a mean of 15.85 oz and a standard deviation of 9.5 oz. A sample of 12 cups is taken. What is the probability that the sample mean is at most 11.25? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #848949 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1987)</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=848949\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> **1. Understand the Problem**\r<BR>\n" );
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document.write( "* We are given the population mean (μ = 15.85 oz) and standard deviation (σ = 0.95 oz) of the amount of soda dispensed by a machine.<BR>\n" );
document.write( "* We have a sample of 12 cups (n = 12).<BR>\n" );
document.write( "* We want to find the probability that the sample mean (x̄) is at most 11.25 oz.\r<BR>\n" );
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document.write( "**2. Assumptions**\r<BR>\n" );
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document.write( "* We assume that the amount of soda dispensed in each cup is normally distributed.<BR>\n" );
document.write( "* We assume that the sample is a random sample.\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 of the mean (σx̄) is calculated as:<BR>\n" );
document.write( "   σx̄ = σ / √n <BR>\n" );
document.write( "   σx̄ = 0.95 / √12 <BR>\n" );
document.write( "   σx̄ ≈ 0.2739 oz\r<BR>\n" );
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document.write( "**4. Standardize the Sample Mean**\r<BR>\n" );
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document.write( "* We need to standardize the sample mean using the z-score formula:<BR>\n" );
document.write( "   z = (x̄ - μ) / σx̄<BR>\n" );
document.write( "   z = (11.25 - 15.85) / 0.2739 <BR>\n" );
document.write( "   z ≈ -16.76\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 ≤ 11.25), which is equivalent to finding P(z ≤ -16.76).<BR>\n" );
document.write( "* Using a standard normal distribution table or a calculator, we find that P(z ≤ -16.76) is extremely close to 0. \r<BR>\n" );
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document.write( "**Conclusion:**\r<BR>\n" );
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document.write( "The probability that the sample mean of 12 cups is at most 11.25 oz is extremely low (approximately 0). This suggests that it is very unlikely to obtain such a low sample mean if the true population mean is 15.85 oz.<BR>\n" );
document.write( " </SPAN>\n" );
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