document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=969825>Question 969825</A>: <I><SPAN STYLE=\"word-break: break-all;\"> 1.Suppose the time to transmit an email message is normally distributed with a mean   <BR>\n" );
document.write( " of 0.72 seconds and a standard deviation of 0.10 seconds.<BR>\n" );
document.write( "a.What is the probability that an email message will require<BR>\n" );
document.write( "i.more than one second to transmit<BR>\n" );
document.write( "ii.between 0.70 and 0.80 second to transmit?\r<BR>\n" );
document.write( "\n" );
document.write( "b.90% of the messages will take at least how many seconds to be transmitted?\r<BR>\n" );
document.write( "\n" );
document.write( "Suppose that a sample of 16 email messages is selected.\r<BR>\n" );
document.write( "\n" );
document.write( "c.What is the probability that the average time for an email message to be transmitted is <BR>\n" );
document.write( "i.more than 0.7 second?<BR>\n" );
document.write( "ii.Between 0.7 and 0.8 second?<BR>\n" );
document.write( "d.90% of the average transmission times will be symmetrically between what two values?\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "2.The average percentage of brown M&M candies in a package of plain M&Ms is 30%. Suppose you randomly select a package of plain M&Ms that contains 55 candies and determine the proportion of brown candies in the package.<BR>\n" );
document.write( "a.What is the probability that the sample proportion of brown candies is less than 20%?<BR>\n" );
document.write( "b.What is the probability that the sample proportion of brown candies is not within 3.5% of the population proportion? <BR>\n" );
document.write( "c.What is the probability that the sample proportion exceeds 35%?<BR>\n" );
document.write( "d.Within what range would you expect the sample proportion to lie symmetrically about 80% of the time?\r<BR>\n" );
document.write( "\n" );
document.write( "*my friends and I had different problems so that I combine the questions that we can't solves this problems. Please help us 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 #592609 by </B><A HREF=/tutors/aboutme.mpl?userid=Boreal><B>Boreal(15235)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Boreal><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=Boreal><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=592609\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> mean= 0.72 s  sd=0.10 s<BR>\n" );
document.write( "z= (1-0.72)/0.10= 0.28/0.10 and it is a one sided z-score of +2.8<BR>\n" );
document.write( "P (z>2.8)=0.0026\r<BR>\n" );
document.write( "\n" );
document.write( "ii.  That would be a z-score of (0.70-0.72)/.10     or z-score of -0.2  P=0.0793 <BR>\n" );
document.write( "     That would be a z-score of (0.8-0.72)/10       or z-score of + 0.8 P=0.2881  ;   Total P=0.3674<BR>\n" );
document.write( "b.  90% messages: would be a z-score of >+1.28   or .128 sec or 0.848 (0.85) sec<BR>\n" );
document.write( "c.  Sampling distribution<BR>\n" );
document.write( "  Probability z < 0.70 sec:  if we get that, we subtract it from 1<BR>\n" );
document.write( "z score is- 0.02/[(0.1)/4]    = 0.08/1  z=-0.8  That probability is 0.2881, so probability > 0.70 sec in sampling distribution is 1-0.2881= 0.7119.\r<BR>\n" );
document.write( "\n" );
document.write( "Between 0.7 sec and 0.8 sec <BR>\n" );
document.write( "(0.7-0.72)/(0.1/4); that is - 0.08/0.1  or z>-0.8    This is 0.2881 on the left side of the curve<BR>\n" );
document.write( "(0.8-0.72)/(0.1/4); that is  +0.08*4/0.1 or z<3.2   This is 0.5000 on the right side<BR>\n" );
document.write( "The probability is 0.7881.\r<BR>\n" );
document.write( "\n" );
document.write( "90% of the average transmission time will be between z values of +/- 1.645<BR>\n" );
document.write( "xb +/- 1.645 (0.1)/4 , where numerical value is 0.041<BR>\n" );
document.write( "Rounding, 90% are between 0.68 and 0.76  for the sample.<BR>\n" );
document.write( "++++++++++++++++++++++++++++++++++++++++++++++++<BR>\n" );
document.write( "2.a z=  (pb-p)/sqrt [(p*(1-p)/55]<BR>\n" );
document.write( "z=(0.2-0.3)/sqrt [(.3*.7)/55]<BR>\n" );
document.write( "-(0.1)/0.0618  =-1.618  This is about 0.055.  \r<BR>\n" );
document.write( "\n" );
document.write( "Not within 3.5% means that sample proportion is >0.335 or <0.265\r<BR>\n" );
document.write( "\n" );
document.write( "Same denominator, numerator is 0.035 <BR>\n" );
document.write( "z value > or < 0.566  This is 0.2857 x2=0.5714\r<BR>\n" );
document.write( "\n" );
document.write( "Same denominator, numerator is now 0.05 and one-way test  shows probability  of 0.21\r<BR>\n" );
document.write( "\n" );
document.write( "80% of the time would be a z-value of +/-1.28 ; I multiply that by the denominator 0.0618 and get 0.079 or about 0.08   The interval would be [0.22, 0.38]\r<BR>\n" );
document.write( "\n" );
document.write( "Note:  This problem is complicated by the fact that n is small;  >35% of the population is 19.25, so greater than 35% starts with 20, which is really 36.3%.  There are issues with this throughout the problem. If you do it on a calculator, with z-scores, you have no problem, but if you input data that would be required to answer the question, you will get different values.<BR>\n" );
document.write( "=========================================\r<BR>\n" );
document.write( "\n" );
document.write( "This was a long pair of problems.  I wouldn't be surprised if I missed something in this, but trying things a couple of ways, by hand and by calculator, I got things to make sense.  <BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );