document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=851656>Question 851656</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Beechwood forests in East Central Europe are<BR>\n" );
document.write( "being threatened by dynamic changes in land ownership and economic upheaval. The<BR>\n" );
document.write( "current status of the beech tree species in this area was evaluated by Hungarian uni-<BR>\n" );
document.write( "versity professors in Applied Ecology and Environmental Research (Vol. 1, 2003). The<BR>\n" );
document.write( "researchers found that 25% of the beech trees in East Central Europe have been damaged<BR>\n" );
document.write( "by fungi. Consider a sample of 20 beech trees from this area.<BR>\n" );
document.write( "1. For any randomly selected beech tree in this area, what is the chance it has been<BR>\n" );
document.write( "damaged by fungi?<BR>\n" );
document.write( "2. Let X denote the number of beech trees, out of the sample, that are damaged by<BR>\n" );
document.write( "fungi. What is the distribution of X? (Specify name and parameter values of<BR>\n" );
document.write( "the X distribution).<BR>\n" );
document.write( "3. What is the variance of X i.e. V(X)?<BR>\n" );
document.write( "4. Use the PMF of X to calculate probability that in the sample there are trees<BR>\n" );
document.write( "damaged by fungi i.e. P(X 6= 0)<BR>\n" );
document.write( "5. Given that there are trees damaged by fungi, what is the probability only one tree<BR>\n" );
document.write( "is damaged? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #512916 by </B><A HREF=/tutors/aboutme.mpl?userid=stanbon><B>stanbon(75887)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=stanbon><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=512916\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Researchers found that 25% of the beech trees in East Central Europe have been damagedby fungi. <BR>\n" );
document.write( "Consider a sample of 20 beech trees from this area.<BR>\n" );
document.write( "1. For any randomly selected beech tree in this area, what is the chance it has beend amaged by fungi?:: Ans: 0.25<BR>\n" );
document.write( "-------------------------------------------------<BR>\n" );
document.write( "2. Let X denote the number of beech trees, out of the sample, that are damaged by fungi. What is the distribution of X? (Specify name and parameter values of<BR>\n" );
document.write( "the X distribution).<BR>\n" );
document.write( "Ans: Binomial because each tree is either fungal of not fungal<BR>\n" );
document.write( "with probability = 0.25 for each tree.<BR>\n" );
document.write( "The mean of each binomial distribution  = np <BR>\n" );
document.write( "In your problem n = 20 and p = 0.25<BR>\n" );
document.write( "So mean = 20(0.25) = 5 <BR>\n" );
document.write( "-----<BR>\n" );
document.write( "The standard deviation of each binomial distribution is sqrt(npq)<BR>\n" );
document.write( "So s = sqrt(npq) = 1.9365<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "3. What is the variance of X i.e. V(X) = s^2 = npq = 3.75<BR>\n" );
document.write( "--------------------------------------<BR>\n" );
document.write( "4. Use the PMF of X to calculate probability that in the sample there are trees<BR>\n" );
document.write( "damaged by fungi i.e. P(X = 6)<BR>\n" );
document.write( "P(X = 6) = 20C6(0.25)^6(0.75)^14 = binompdf(20,0.25,6) = 0.1686<BR>\n" );
document.write( "-----------------------------------------------<BR>\n" );
document.write( "5. Given that there are  trees damaged by fungi, what is the probability only one tree is damaged?<BR>\n" );
document.write( "Question?::  P(x = 1) = 20C1(0.25)*(0.75)^19 = binompdf(20,0.25,1) = 0.0211<BR>\n" );
document.write( "==============================<BR>\n" );
document.write( "Cheers,<BR>\n" );
document.write( "Stan H.<BR>\n" );
document.write( "================== </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );