document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=380422>Question 380422</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Let the random variable X follow a normal distribution with mean &#956; = 50 and<BR>\n" );
document.write( "variance &#963;2 = 25.<BR>\n" );
document.write( "a) Find the probability that X is greater than 58.<BR>\n" );
document.write( "b) Find the probability that X is greater than 46 and less than 60.<BR>\n" );
document.write( "c) If you draw a sample of size n=20 from the X population described above,<BR>\n" );
document.write( "what is the probability that the sample variance, s2, is larger than 30?<BR>\n" );
document.write( "d) Imagine a symmetric interval about the mean (&#956;ąc) of the distribution<BR>\n" );
document.write( "described above. Find the value for c such that the probability is<BR>\n" );
document.write( "approximately 0.1 that X falls in this interval. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #269981 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=269981\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\">  Let the random variable X follow a normal distribution with mean &#956; = 50 and<BR>\n" );
document.write( "variance &#963;2 = 25.<BR>\n" );
document.write( "Note: std = 5<BR>\n" );
document.write( "=======================<BR>\n" );
document.write( "a) Find the probability that X is greater than 58.<BR>\n" );
document.write( "z(58) = (58-50)/5 = 8/5 = 1.4<BR>\n" );
document.write( "P(x> 58) = P(z> 1.4) = <BR>\n" );
document.write( "--------------------------------<BR>\n" );
document.write( "b) Find the probability that X is greater than 46 and less than 60.<BR>\n" );
document.write( "Do the same as above for 46 and 60.<BR>\n" );
document.write( "------------------------------------------------<BR>\n" );
document.write( "c) If you draw a sample of size n=20 from the X population described above,<BR>\n" );
document.write( "what is the probability that the sample variance, s2, is larger than 30?<BR>\n" );
document.write( "std of the sample means = 5/sqrt(20)<BR>\n" );
document.write( "---<BR>\n" );
document.write( "t(30) = (30-50)/[5/sqrt(20)] = -17.89<BR>\n" );
document.write( "----<BR>\n" );
document.write( "P(x-bar > 30) = P(t > -17.89 when df = 19) = tcdf(-17.89,100,19) = 1.00<BR>\n" );
document.write( "----------------------------<BR>\n" );
document.write( "d) Imagine a symmetric interval about the mean (&#956;ąc) of the distribution<BR>\n" );
document.write( "described above. Find the value for c such that the probability is<BR>\n" );
document.write( "approximately 0.1 that X falls in this interval.<BR>\n" );
document.write( "----<BR>\n" );
document.write( "Draw the picture using a normal curve.<BR>\n" );
document.write( "You have an interval with 0.05 to the left and to the right of the mean.<BR>\n" );
document.write( "That leaves a left-tail of 0.45 to the left.<BR>\n" );
document.write( "z = +/-invNorm(0.45) = +-0.1257<BR>\n" );
document.write( "---------------<BR>\n" );
document.write( "Left boundary:<BR>\n" );
document.write( "c = -0.1257*5+50 = 49.37<BR>\n" );
document.write( "---<BR>\n" );
document.write( "Right boundary:<BR>\n" );
document.write( "c = +0.1257*5+50 = 50.63<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" );