document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1194732>Question 1194732</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Each minute a machine produces a length of rope with<BR>\n" );
document.write( "mean of 4 feet and standard deviation of 5 inches. Assuming that the <BR>\n" );
document.write( "amounts produced in different minutes are independent and <BR>\n" );
document.write( "identically distributed, approximate the probability that the machine will produce at least <BR>\n" );
document.write( "250 feet in one hour. (Use the central limit theorem, ∅ = (0.26) 0.6026<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 #848444 by </B><A HREF=/tutors/aboutme.mpl?userid=ElectricPavlov><B>ElectricPavlov(122)</B></A><img src=\"/images/stars/stars-09.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=ElectricPavlov><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=848444\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> **1. Define Variables**\r<BR>\n" );
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document.write( "* Let X be the length of rope produced in one minute.<BR>\n" );
document.write( "* Let S be the total length of rope produced in one hour (60 minutes).\r<BR>\n" );
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document.write( "**2. Given**\r<BR>\n" );
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document.write( "* Mean length of rope per minute (μ_X) = 4 feet<BR>\n" );
document.write( "* Standard deviation of rope per minute (σ_X) = 5 inches = 5/12 feet <BR>\n" );
document.write( "* Number of minutes in an hour (n) = 60\r<BR>\n" );
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document.write( "**3. Apply Central Limit Theorem**\r<BR>\n" );
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document.write( "* Since S is the sum of 60 independent and identically distributed random variables (X), the Central Limit Theorem states that the distribution of S will be approximately normal.\r<BR>\n" );
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document.write( "* **Mean of Total Length (μ_S):**<BR>\n" );
document.write( "    * μ_S = n * μ_X = 60 minutes * 4 feet/minute = 240 feet\r<BR>\n" );
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document.write( "* **Standard Deviation of Total Length (σ_S):**<BR>\n" );
document.write( "    * σ_S = √(n) * σ_X = √(60) * (5/12) feet ≈ 3.229 feet\r<BR>\n" );
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document.write( "**4. Standardize the Value**\r<BR>\n" );
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document.write( "* We want to find P(S ≥ 250 feet)<BR>\n" );
document.write( "* Standardize 250 feet:<BR>\n" );
document.write( "    * z = (X - μ_S) / σ_S <BR>\n" );
document.write( "    * z = (250 feet - 240 feet) / 3.229 feet <BR>\n" );
document.write( "    * z ≈ 3.09\r<BR>\n" );
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document.write( "**5. Find the Probability**\r<BR>\n" );
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document.write( "* Using a standard normal distribution table or a calculator:<BR>\n" );
document.write( "    * P(S ≥ 250) = P(Z ≥ 3.09) ≈ 0.001\r<BR>\n" );
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document.write( "**Therefore, the approximate probability that the machine will produce at least 250 feet of rope in one hour is 0.001 (or 0.1%).**\r<BR>\n" );
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document.write( "**Key Points:**\r<BR>\n" );
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document.write( "* The Central Limit Theorem allows us to approximate the distribution of the sum of a large number of independent and identically distributed random variables as normal, even if the individual variables themselves are not normally distributed.<BR>\n" );
document.write( "* In this case, we assumed that the length of rope produced in each minute is independent of the length produced in other minutes.<BR>\n" );
document.write( "* The calculation involves standardizing the value of interest (250 feet) and then using the standard normal distribution to find the corresponding probability.<BR>\n" );
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