document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1172252>Question 1172252</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The relative frequencies of blood lead concentrations for two groups of workers in Canada-one<BR>\n" );
document.write( "examined in 1979 and the other in 1987-are displayed below.<BR>\n" );
document.write( "Blood Lead<BR>\n" );
document.write( "(µg/dl)<BR>\n" );
document.write( "1979<BR>\n" );
document.write( "(%)<BR>\n" );
document.write( "1987<BR>\n" );
document.write( "(%)<BR>\n" );
document.write( "<20 11.5 37.8<BR>\n" );
document.write( "20-29 12.1 14.7<BR>\n" );
document.write( "30-39 13.9 13.1<BR>\n" );
document.write( "40-49 15.4 15.3<BR>\n" );
document.write( "50-59 16.5 10.5<BR>\n" );
document.write( "60-69 12.8 6.8<BR>\n" );
document.write( "70-79 8.4 1.4<BR>\n" );
document.write( " 80 9.4 0.4<BR>\n" );
document.write( "a) In which of the two years do the workers tend to have lower blood lead levels?<BR>\n" );
document.write( "b) Compute the cumulative relative frequencies for each group of workers. Use these data<BR>\n" );
document.write( "to construct a pair of cumulative frequency polygons.<BR>\n" );
document.write( "c) For which group of workers is the distribution of blood lead levels stochastically larger? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #850827 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=850827\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Absolutely! Let's analyze the blood lead concentration data.\r<BR>\n" );
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document.write( "**a) Lower Blood Lead Levels**\r<BR>\n" );
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document.write( "* By observing the first row of data, we can see that in 1987, 37.8% of workers had blood lead levels below 20 µg/dl, while in 1979, only 11.5% had levels that low.<BR>\n" );
document.write( "* Also, as we go down the table, the percentages for higher lead levels are much lower in 1987.<BR>\n" );
document.write( "* Therefore, the workers in **1987** tend to have lower blood lead levels.\r<BR>\n" );
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document.write( "**b) Cumulative Relative Frequencies and Cumulative Frequency Polygons**\r<BR>\n" );
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document.write( "* **Cumulative Relative Frequencies:**<BR>\n" );
document.write( "    * **1979:**<BR>\n" );
document.write( "        * <20: 11.5%<BR>\n" );
document.write( "        * <30: 11.5 + 12.1 = 23.6%<BR>\n" );
document.write( "        * <40: 23.6 + 13.9 = 37.5%<BR>\n" );
document.write( "        * <50: 37.5 + 15.4 = 52.9%<BR>\n" );
document.write( "        * <60: 52.9 + 16.5 = 69.4%<BR>\n" );
document.write( "        * <70: 69.4 + 12.8 = 82.2%<BR>\n" );
document.write( "        * <80: 82.2 + 8.4 = 90.6%<BR>\n" );
document.write( "        * ≥ 80: 90.6 + 9.4 = 100%<BR>\n" );
document.write( "    * **1987:**<BR>\n" );
document.write( "        * <20: 37.8%<BR>\n" );
document.write( "        * <30: 37.8 + 14.7 = 52.5%<BR>\n" );
document.write( "        * <40: 52.5 + 13.1 = 65.6%<BR>\n" );
document.write( "        * <50: 65.6 + 15.3 = 80.9%<BR>\n" );
document.write( "        * <60: 80.9 + 10.5 = 91.4%<BR>\n" );
document.write( "        * <70: 91.4 + 6.8 = 98.2%<BR>\n" );
document.write( "        * <80: 98.2 + 1.4 = 99.6%<BR>\n" );
document.write( "        * ≥ 80: 99.6 + 0.4 = 100%\r<BR>\n" );
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document.write( "* **Cumulative Frequency Polygons:**<BR>\n" );
document.write( "    * To construct the polygons, plot the cumulative frequencies against the upper limit of each blood lead concentration interval.<BR>\n" );
document.write( "    * The 1987 polygon will be shifted to the left of the 1979 polygon, visually indicating lower blood lead levels.\r<BR>\n" );
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document.write( "**c) Stochastic Dominance**\r<BR>\n" );
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document.write( "* A distribution is stochastically larger if its cumulative distribution function is always below or equal to the other distribution's cumulative distribution function.<BR>\n" );
document.write( "* In this case, the cumulative relative frequencies for 1987 are consistently higher than those for 1979.<BR>\n" );
document.write( "* Therefore, the **1987** group of workers has a stochastically larger distribution of blood lead levels. This indicates that the 1987 group has a greater probability of having lower blood lead levels.<BR>\n" );
document.write( " </SPAN>\n" );
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