document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1198232>Question 1198232</A>: <I><SPAN STYLE=\"word-break: break-all;\"> It takes an average of 40 seconds to download a certain file, with a standard deviation of 5 seconds. The actual distribution of the download time is unknown. Using Chebyshev’s inequality, what can be said about the probability of spending more than 1 minute for this download? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #848487 by </B><A HREF=/tutors/aboutme.mpl?userid=proyaop><B>proyaop(69)</B></A><img src=\"/images/stars/stars-08.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=proyaop><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=848487\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> **1. Define the Problem**\r<BR>\n" );
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document.write( "* We want to find the probability of spending more than 1 minute (60 seconds) to download the file.<BR>\n" );
document.write( "* Mean download time (μ) = 40 seconds<BR>\n" );
document.write( "* Standard deviation (σ) = 5 seconds\r<BR>\n" );
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document.write( "**2. Calculate the Number of Standard Deviations**\r<BR>\n" );
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document.write( "* Difference from the mean: 60 seconds - 40 seconds = 20 seconds<BR>\n" );
document.write( "* Number of standard deviations (k): 20 seconds / 5 seconds = 4\r<BR>\n" );
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document.write( "**3. Apply Chebyshev's Inequality**\r<BR>\n" );
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document.write( "* Chebyshev's Inequality states that for any dataset, the proportion of data that lies within 'k' standard deviations of the mean is at least 1 - (1/k²).\r<BR>\n" );
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document.write( "* In this case, k = 4.\r<BR>\n" );
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document.write( "* Probability of data within 4 standard deviations of the mean: <BR>\n" );
document.write( "    * 1 - (1/4²) = 1 - (1/16) = 15/16 = 0.9375\r<BR>\n" );
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document.write( "* This means at least 93.75% of the download times will fall within 4 standard deviations of the mean (between 0 and 80 seconds).\r<BR>\n" );
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document.write( "**4. Determine the Probability of Spending More Than 1 Minute**\r<BR>\n" );
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document.write( "* Since we want the probability of spending *more* than 1 minute (60 seconds), we are looking at the probability of values outside of 4 standard deviations from the mean.\r<BR>\n" );
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document.write( "* Probability of spending more than 1 minute: <BR>\n" );
document.write( "    * 1 - Probability of spending within 4 standard deviations <BR>\n" );
document.write( "    * 1 - 0.9375 = 0.0625\r<BR>\n" );
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document.write( "**Therefore, according to Chebyshev's Inequality, the probability of spending more than 1 minute to download the file is at most 0.0625 or 6.25%.**\r<BR>\n" );
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document.write( "**Important Note:**\r<BR>\n" );
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document.write( "* Chebyshev's Inequality provides an upper bound on the probability. The actual probability could be lower. <BR>\n" );
document.write( "* If the distribution of download times were known (e.g., normal distribution), we could use more precise methods (like the z-score table) to calculate the exact probability.<BR>\n" );
document.write( " </SPAN>\n" );
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