document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1192139>Question 1192139</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Assume that the probability of a being born with Genetic Condition B is `pi=`7/20``. A study looks at a random sample of 423 volunteers.\r<BR>\n" );
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document.write( "Find the most likely number of the 423 volunteers to have Genetic Condition B.<BR>\n" );
document.write( "(Round answer to one decimal place.)<BR>\n" );
document.write( "μ = \r<BR>\n" );
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document.write( "Let `X` represent the number of volunteers (out of 423) who have Genetic Condition B. Find the standard deviation for the probability distribution of `X`.<BR>\n" );
document.write( "(Round answer to two decimal places.)<BR>\n" );
document.write( "σ = \r<BR>\n" );
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document.write( "Use the range rule of thumb to find the minimum usual value μ  -  2σ and the maximum usual value μ+2σ.<BR>\n" );
document.write( "Enter answer as an interval using square-brackets only with whole numbers.<BR>\n" );
document.write( "usual values = <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 #849078 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=849078\">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. Most Likely Number with Genetic Condition B:**\r<BR>\n" );
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document.write( "The most likely number of volunteers with Genetic Condition B is simply the expected value (mean) of the distribution.  For a binomial distribution (which this is, since each volunteer either has the condition or doesn't), the mean is:\r<BR>\n" );
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document.write( "μ = n * p\r<BR>\n" );
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document.write( "Where:\r<BR>\n" );
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document.write( "* n = sample size = 423<BR>\n" );
document.write( "* p = probability of having the condition = 7/20 = 0.35\r<BR>\n" );
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document.write( "μ = 423 * 0.35 = 148.05\r<BR>\n" );
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document.write( "Rounding to one decimal place, the most likely number is 148.1.\r<BR>\n" );
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document.write( "**2. Standard Deviation:**\r<BR>\n" );
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document.write( "The standard deviation for a binomial distribution is:\r<BR>\n" );
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document.write( "σ = √(n * p * (1 - p))\r<BR>\n" );
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document.write( "σ = √(423 * 0.35 * (1 - 0.35))<BR>\n" );
document.write( "σ = √(423 * 0.35 * 0.65)<BR>\n" );
document.write( "σ = √96.1575<BR>\n" );
document.write( "σ ≈ 9.81\r<BR>\n" );
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document.write( "Rounding to two decimal places, the standard deviation is 9.81.\r<BR>\n" );
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document.write( "**3. Range Rule of Thumb:**\r<BR>\n" );
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document.write( "The range rule of thumb states that most values fall within two standard deviations of the mean.\r<BR>\n" );
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document.write( "* **Minimum usual value:** μ - 2σ = 148.05 - 2 * 9.81 = 148.05 - 19.62 = 128.43<BR>\n" );
document.write( "* **Maximum usual value:** μ + 2σ = 148.05 + 2 * 9.81 = 148.05 + 19.62 = 167.67\r<BR>\n" );
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document.write( "Since we're dealing with whole numbers (number of volunteers), we round these to the nearest whole number.\r<BR>\n" );
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document.write( "Therefore, the usual values are [128, 168].<BR>\n" );
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