document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1199757>Question 1199757</A>: <I><SPAN STYLE=\"word-break: break-all;\"> 8% of all Americans live in poverty. If 37 Americans are randomly selected, find the probability that\r<BR>\n" );
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document.write( "a.  Exactly 3 of them live in poverty. \r<BR>\n" );
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document.write( "b.  At most 5 of them live in poverty. \r<BR>\n" );
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document.write( "c.  At least 4 of them live in poverty. \r<BR>\n" );
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document.write( "d.  Between 2 and 6 (including 2 and 6) of them live in poverty. <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 #848283 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=848283\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> **a) Probability of exactly 3 of them live in poverty.**\r<BR>\n" );
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document.write( "* This is a binomial probability problem. <BR>\n" );
document.write( "* n = 37 (sample size)<BR>\n" );
document.write( "* p = 0.08 (probability of success - living in poverty)<BR>\n" );
document.write( "* q = 1 - p = 0.92 (probability of failure - not living in poverty)<BR>\n" );
document.write( "* x = 3 (number of successes)\r<BR>\n" );
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document.write( "* P(X = 3) = (37C3) * (0.08)^3 * (0.92)^(37-3) <BR>\n" );
document.write( "   * where (37C3) is the number of combinations of 37 things taken 3 at a time.\r<BR>\n" );
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document.write( "* Using a binomial probability calculator or statistical software:<BR>\n" );
document.write( "   P(X = 3) ≈ 0.2079\r<BR>\n" );
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document.write( "**b) Probability of at most 5 of them live in poverty.**\r<BR>\n" );
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document.write( "* P(X ≤ 5) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4) + P(X = 5)\r<BR>\n" );
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document.write( "* Use a cumulative binomial probability calculator or statistical software:<BR>\n" );
document.write( "   P(X ≤ 5) ≈ 0.9589\r<BR>\n" );
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document.write( "**c) Probability of at least 4 of them live in poverty.**\r<BR>\n" );
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document.write( "* P(X ≥ 4) = P(X = 4) + P(X = 5) + ... + P(X = 37)\r<BR>\n" );
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document.write( "* Alternatively:<BR>\n" );
document.write( "   P(X ≥ 4) = 1 - P(X ≤ 3) <BR>\n" );
document.write( "   P(X ≥ 4) = 1 - (P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3)) <BR>\n" );
document.write( "   P(X ≥ 4) ≈ 1 - 0.7847 = 0.2153\r<BR>\n" );
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document.write( "**d) Probability of between 2 and 6 of them live in poverty (inclusive):**\r<BR>\n" );
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document.write( "* P(2 ≤ X ≤ 6) = P(X = 2) + P(X = 3) + P(X = 4) + P(X = 5) + P(X = 6) \r<BR>\n" );
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document.write( "* Use a cumulative binomial probability calculator or statistical software to find:<BR>\n" );
document.write( "   P(2 ≤ X ≤ 6) ≈ 0.9983\r<BR>\n" );
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document.write( "**In summary:**\r<BR>\n" );
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document.write( "* a) P(X = 3) ≈ 0.2079<BR>\n" );
document.write( "* b) P(X ≤ 5) ≈ 0.9589<BR>\n" );
document.write( "* c) P(X ≥ 4) ≈ 0.2153<BR>\n" );
document.write( "* d) P(2 ≤ X ≤ 6) ≈ 0.9983<BR>\n" );
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