document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1161365>Question 1161365</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Given the following table:<BR>\n" );
document.write( "y   | 15  30  60  90<BR>\n" );
document.write( "p(y)| .2  .3  .4  .1\r<BR>\n" );
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document.write( "Find: Mean, Standard deviation, and probability that the time is within 1 SD of its mean.\r<BR>\n" );
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document.write( "Any help would be appreciated, thanks! </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #784891 by </B><A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><B>jim_thompson5910(35256)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><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=784891\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "Answers:<BR>\n" );
document.write( "Mean = <font color=red size=4>45</font><BR>\n" );
document.write( "Standard deviation = <font color=red size=4>23.2379000772446</font> (approximate; round however you need to)<BR>\n" );
document.write( "Probability the time is within one standard deviation of the mean = <font color=red size=4>0.7</font>\r<BR>\n" );
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document.write( "Work Shown:\r<BR>\n" );
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document.write( "Part (a): Finding the mean\r<BR>\n" );
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document.write( "To find the mean, we multiply each y value by its corresponding probability p(y) value as shown in the table below<BR>\n" );
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document.write( "<tr><TD >y</TD><TD >p(y)</TD><TD >y*p(y)</TD></tr><tr><TD >15</TD><TD >0.2</TD><TD >3</TD></tr><tr><TD >30</TD><TD >0.3</TD><TD >9</TD></tr><tr><TD >60</TD><TD >0.4</TD><TD >24</TD></tr><tr><TD >90</TD><TD >0.1</TD><TD >9</TD></tr>\n" );
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document.write( "Example: in row 1 we have y*p(y) = 15*0.2 = 3\r<BR>\n" );
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document.write( "Add up the values in the y*p(y) column to get the mean<BR>\n" );
document.write( "mean = 3+9+24+9 = 45\r<BR>\n" );
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document.write( "We will use the mean later when it comes to calculating the standard deviation\r<BR>\n" );
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document.write( "Part (b): Finding the standard deviation\r<BR>\n" );
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document.write( "First we must construct a new column labeled y^2*p(y). This is a column of values where we square the y values, and then multiply those squares by their corresponding p(y) values. For instance, row 1 has y = 15, so y^2 = 15^2 = 225, which leads to y^2*p(y) = 225*0.2 = 45. It's a coicidence that this also happens to be the value of the mean. The other rows are computed in a similar fashion.\r<BR>\n" );
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document.write( "Here's the updated table<BR>\n" );
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document.write( "<tr><TD >y</TD><TD >p(y)</TD><TD >y*p(y)</TD><TD >y^2*p(y)</TD></tr><tr><TD >15</TD><TD >0.2</TD><TD >3</TD><TD >45</TD></tr><tr><TD >30</TD><TD >0.3</TD><TD >9</TD><TD >270</TD></tr><tr><TD >60</TD><TD >0.4</TD><TD >24</TD><TD >1440</TD></tr><tr><TD >90</TD><TD >0.1</TD><TD >9</TD><TD >810</TD></tr>\n" );
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document.write( "We then add up everything in that new y^2*p(y) column which yields<BR>\n" );
document.write( "45+270+1440+810 = 2565\r<BR>\n" );
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document.write( "Next, we square the mean.<BR>\n" );
document.write( "mu = mean<BR>\n" );
document.write( "mu = 45 found earlier<BR>\n" );
document.write( "mu^2 = 45^2<BR>\n" );
document.write( "mu^2 = 2025\r<BR>\n" );
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document.write( "Subtract this result off the sum of the y^2*p(y) column to get the variance sigma^2<BR>\n" );
document.write( "sigma^2 = (sum of y*p(y) column) - ( mu^2 )<BR>\n" );
document.write( "sigma^2 = 2565 - 2025<BR>\n" );
document.write( "sigma^2 = 540\r<BR>\n" );
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document.write( "Lastly, we apply the square root to both sides to get the value of sigma, which is the standard deviation<BR>\n" );
document.write( "sigma^2 = 540<BR>\n" );
document.write( "sigma = sqrt(540)<BR>\n" );
document.write( "sigma = 23.2379000772446 which is approximate\r<BR>\n" );
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document.write( "We will use this for part (c), along with the mean as well. \r<BR>\n" );
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document.write( "Part (c): probability that the time is within 1 SD of its mean\r<BR>\n" );
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document.write( "From part (b), we found the approximate standard deviation to be 23.2379000772446\r<BR>\n" );
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document.write( "Add and subtract this from the mean to get the values of U and L respectively<BR>\n" );
document.write( "L = lower value that is 1 std dev from the mean<BR>\n" );
document.write( "L = mu-sigma<BR>\n" );
document.write( "L = 45 - 23.2379000772446<BR>\n" );
document.write( "L = 21.7620999227553 which is approximate\r<BR>\n" );
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document.write( "U = upper value that is 1 std dev from the mean<BR>\n" );
document.write( "U = mu+sigma<BR>\n" );
document.write( "U = 45 + 23.2379000772446<BR>\n" );
document.write( "U = 68.2379000772447\r<BR>\n" );
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document.write( "The values of L and U were approximately 21.7620999227553 and 68.2379000772447\r<BR>\n" );
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document.write( "Looking at the table, we are searching for y values that are between those L and U values.<BR>\n" );
document.write( "This applies for y = 30 and y = 60 only<BR>\n" );
document.write( "In other words, y = 30 and y = 60 make the inequality 21.7620999227553 < y < 68.2379000772447 true.\r<BR>\n" );
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document.write( "So from here, we add up the corresponding probabilities for those y values<BR>\n" );
document.write( "0.3 + 0.4 = 0.7\r<BR>\n" );
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