document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1178064>Question 1178064</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Let X be a random variable that represents the diastolic blood pressure (DBP) of the population of 18- to 74-year-old men in the United States who are not taking any corrective medication. Suppose that X has mean 80.7 mm Hg and standard deviation 9.2.<BR>\n" );
document.write( "(a) Obtain a bound on the probability that the DBP of this population will assumes values<BR>\n" );
document.write( "between 53.1 and 108.3 mm Hg.<BR>\n" );
document.write( "(b) In addition, assume that the distribution of X is approximately normal. Using the<BR>\n" );
document.write( "normal table, find P(53.1 ≤ X ≤ 108.3). Compare this with the empirical rule.\r<BR>\n" );
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document.write( "Thank you :) </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #850365 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=850365\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let's solve this problem step-by-step.\r<BR>\n" );
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document.write( "**Given Information**\r<BR>\n" );
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document.write( "* X: Diastolic blood pressure (DBP)<BR>\n" );
document.write( "* μ (mean) = 80.7 mm Hg<BR>\n" );
document.write( "* σ (standard deviation) = 9.2 mm Hg\r<BR>\n" );
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document.write( "**(a) Chebyshev's Inequality**\r<BR>\n" );
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document.write( "Chebyshev's Inequality provides a bound on the probability that a random variable falls within a certain range. It states:\r<BR>\n" );
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document.write( "* P(|X - μ| ≥ kσ) ≤ 1/k²<BR>\n" );
document.write( "* Or equivalently, P(|X - μ| < kσ) ≥ 1 - 1/k²\r<BR>\n" );
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document.write( "We want to find P(53.1 ≤ X ≤ 108.3). Let's rewrite this as:\r<BR>\n" );
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document.write( "* |X - μ| < kσ<BR>\n" );
document.write( "* |X - 80.7| < k(9.2)\r<BR>\n" );
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document.write( "We need to find the range of X:\r<BR>\n" );
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document.write( "* 108.3 - 80.7 = 27.6<BR>\n" );
document.write( "* 80.7 - 53.1 = 27.6\r<BR>\n" );
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document.write( "So, we have:\r<BR>\n" );
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document.write( "* |X - 80.7| < 27.6\r<BR>\n" );
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document.write( "Now, find k:\r<BR>\n" );
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document.write( "* k(9.2) = 27.6<BR>\n" );
document.write( "* k = 27.6 / 9.2 = 3\r<BR>\n" );
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document.write( "Now, apply Chebyshev's Inequality:\r<BR>\n" );
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document.write( "* P(|X - 80.7| < 27.6) ≥ 1 - 1/k²<BR>\n" );
document.write( "* P(53.1 ≤ X ≤ 108.3) ≥ 1 - 1/3²<BR>\n" );
document.write( "* P(53.1 ≤ X ≤ 108.3) ≥ 1 - 1/9 = 8/9 ≈ 0.8889\r<BR>\n" );
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document.write( "Therefore, a bound on the probability is 8/9 or approximately 0.8889.\r<BR>\n" );
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document.write( "**(b) Normal Distribution and Empirical Rule**\r<BR>\n" );
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document.write( "Assume X is normally distributed.\r<BR>\n" );
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document.write( "1.  **Calculate Z-scores:**<BR>\n" );
document.write( "    * Z1 = (53.1 - 80.7) / 9.2 = -27.6 / 9.2 = -3<BR>\n" );
document.write( "    * Z2 = (108.3 - 80.7) / 9.2 = 27.6 / 9.2 = 3\r<BR>\n" );
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document.write( "2.  **Find P(-3 ≤ Z ≤ 3) using Normal Table:**<BR>\n" );
document.write( "    * P(-3 ≤ Z ≤ 3) = P(Z ≤ 3) - P(Z ≤ -3)<BR>\n" );
document.write( "    * From the normal table, P(Z ≤ 3) ≈ 0.9987 and P(Z ≤ -3) ≈ 0.0013<BR>\n" );
document.write( "    * P(-3 ≤ Z ≤ 3) = 0.9987 - 0.0013 = 0.9974\r<BR>\n" );
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document.write( "3.  **Empirical Rule (68-95-99.7 Rule):**<BR>\n" );
document.write( "    * The empirical rule states that approximately 99.7% of the data falls within 3 standard deviations of the mean in a normal distribution.<BR>\n" );
document.write( "    * This is consistent with our calculated value of 0.9974.\r<BR>\n" );
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document.write( "**Comparison**\r<BR>\n" );
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document.write( "* **Chebyshev's Inequality:** Provides a lower bound (8/9 ≈ 0.8889). It is a general result and works for any distribution with a defined mean and standard deviation.<BR>\n" );
document.write( "* **Normal Distribution:** Provides a more precise probability (0.9974) when the distribution is known to be normal.<BR>\n" );
document.write( "* **Empirical Rule:** Agrees with the Normal Distribution calculation.\r<BR>\n" );
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document.write( "**Results**\r<BR>\n" );
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document.write( "(a) A bound on the probability is 8/9 or approximately 0.8889.<BR>\n" );
document.write( "(b) P(53.1 ≤ X ≤ 108.3) ≈ 0.9974. This agrees with the empirical rule.<BR>\n" );
document.write( " </SPAN>\n" );
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