document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1202369>Question 1202369</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The following scores represent the final examination grades for an elementary <BR>\n" );
document.write( "statistics course: <BR>\n" );
document.write( "23 60 79 32 57 74 52 70 82 36 <BR>\n" );
document.write( "80 77 81 95 41 65 92 85 55 76 <BR>\n" );
document.write( "52 10 64 75 78 25 80 98 81 67 <BR>\n" );
document.write( "41 71 83 54 64 72 88 62 74 43 <BR>\n" );
document.write( "60 78 89 76 84 48 84 90 15 79 <BR>\n" );
document.write( "34 67 17 82 69 74 63 80 85 61 <BR>\n" );
document.write( "a) Make a stem-and-leaf display for these data. Show at least five stems. Sort <BR>\n" );
document.write( "the data on each stem in order of increasing magnitude. <BR>\n" );
document.write( "b) Find the median, lower quartile, and upper quartile of these data. <BR>\n" );
document.write( "c) What fraction of the class received scores which were less than 65? <BR>\n" );
document.write( "d) Make a frequency table, starting the first class interval at a lower class <BR>\n" );
document.write( "boundary of 9.5. Use Sturges’ Rule.\r<BR>\n" );
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document.write( "e) Draw a frequency histogram.\r<BR>\n" );
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document.write( "f) Draw a relative frequency histogram on the same x-axis.\r<BR>\n" );
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document.write( "g) Draw a cumulative frequency diagram.\r<BR>\n" );
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document.write( "h) Find the mode.\r<BR>\n" );
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document.write( "i) Show a box plot of these data. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #837158 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3817)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><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=837158\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "I'll do the first three parts to get you started.\r<BR>\n" );
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document.write( "Part (a)\r<BR>\n" );
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document.write( "Given data set<BR>\n" );
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document.write( "<tr><TD >23</TD><TD >60</TD><TD >79</TD><TD >32</TD><TD >57</TD><TD >74</TD><TD >52</TD><TD >70</TD><TD >82</TD><TD >36</TD></tr><tr><TD >80</TD><TD >77</TD><TD >81</TD><TD >95</TD><TD >41</TD><TD >65</TD><TD >92</TD><TD >85</TD><TD >55</TD><TD >76</TD></tr><tr><TD >52</TD><TD >10</TD><TD >64</TD><TD >75</TD><TD >78</TD><TD >25</TD><TD >80</TD><TD >98</TD><TD >81</TD><TD >67</TD></tr><tr><TD >41</TD><TD >71</TD><TD >83</TD><TD >54</TD><TD >64</TD><TD >72</TD><TD >88</TD><TD >62</TD><TD >74</TD><TD >43</TD></tr><tr><TD >60</TD><TD >78</TD><TD >89</TD><TD >76</TD><TD >84</TD><TD >48</TD><TD >84</TD><TD >90</TD><TD >15</TD><TD >79</TD></tr><tr><TD >34</TD><TD >67</TD><TD >17</TD><TD >82</TD><TD >69</TD><TD >74</TD><TD >63</TD><TD >80</TD><TD >85</TD><TD >61</TD></tr>\n" );
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document.write( "There are 6 rows and 10 columns to give 6*10 = 60 values total.\r<BR>\n" );
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document.write( "Let's sort the data from smallest to largest. Use a spreadsheet to do this quickly.<BR>\n" );
document.write( "<TABLE  border = \"1\" cellpadding = \"5\">\n" );
document.write( "<tr><TD >10</TD><TD >15</TD><TD >17</TD><TD >23</TD><TD >25</TD><TD >32</TD><TD >34</TD><TD >36</TD><TD >41</TD><TD >41</TD></tr><tr><TD >43</TD><TD >48</TD><TD >52</TD><TD >52</TD><TD >54</TD><TD >55</TD><TD >57</TD><TD >60</TD><TD >60</TD><TD >61</TD></tr><tr><TD >62</TD><TD >63</TD><TD >64</TD><TD >64</TD><TD >65</TD><TD >67</TD><TD >67</TD><TD >69</TD><TD >70</TD><TD >71</TD></tr><tr><TD >72</TD><TD >74</TD><TD >74</TD><TD >74</TD><TD >75</TD><TD >76</TD><TD >76</TD><TD >77</TD><TD >78</TD><TD >78</TD></tr><tr><TD >79</TD><TD >79</TD><TD >80</TD><TD >80</TD><TD >80</TD><TD >81</TD><TD >81</TD><TD >82</TD><TD >82</TD><TD >83</TD></tr><tr><TD >84</TD><TD >84</TD><TD >85</TD><TD >85</TD><TD >88</TD><TD >89</TD><TD >90</TD><TD >92</TD><TD >95</TD><TD >98</TD></tr>\n" );
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document.write( "From here, let's break up the data values so that each line has the same tens place value.<BR>\n" );
document.write( "<ul><li>10,15,17,</li><li>23,25,</li><li>32,34,36,</li><li>41,41,43,48</li><li>52,52,54,55,57,</li><li>60,60,61,62,63,64,64,65,67,67,69,</li><li>70,71,72,74,74,74,75,76,76,77,78,78,79,79</li><li>80,80,80,81,81,82,82,83,84,84,85,85,88,89,</li><li>90,92,95,98</li></ul>Then each stem is a different tens value. The stems are: 1,2,3,4,5,6,7,8,9<BR>\n" );
document.write( "They represent 10,20,30,40,50,60,70,80,90 respectively.\r<BR>\n" );
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document.write( "<tr><TD >Stem</TD><TD >Leaf</TD></tr><tr><TD >1</TD><TD ></TD></tr><tr><TD >2</TD><TD ></TD></tr><tr><TD >3</TD><TD ></TD></tr><tr><TD >4</TD><TD ></TD></tr><tr><TD >5</TD><TD ></TD></tr><tr><TD >6</TD><TD ></TD></tr><tr><TD >7</TD><TD ></TD></tr><tr><TD >8</TD><TD ></TD></tr><tr><TD >9</TD><TD ></TD></tr>\n" );
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document.write( "To fill out the remaining entries, we will write the units digit of each value. <BR>\n" );
document.write( "Examples: <BR>\n" );
document.write( "The value 10 has stem 1 and leaf 0<BR>\n" );
document.write( "The value 15 has stem 1 and leaf 5<BR>\n" );
document.write( "The value 17 has stem 1 and leaf 7\r<BR>\n" );
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document.write( "I'll separate each leaf by a comma. Some textbooks will use a space as the separater. Other times there isn't a separater at all (and all of the leaves on one line are crammed together as one long string of numbers)<BR>\n" );
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document.write( "<tr><TD >Stem</TD><TD >Leaf</TD></tr><tr><TD >1</TD><TD >0,5,7</TD></tr><tr><TD >2</TD><TD >3,5</TD></tr><tr><TD >3</TD><TD >2,4,6</TD></tr><tr><TD >4</TD><TD >1,1,3,8</TD></tr><tr><TD >5</TD><TD >2,2,4,5,7</TD></tr><tr><TD >6</TD><TD >0,0,1,2,3,4,4,5,7,7,9</TD></tr><tr><TD >7</TD><TD >0,1,2,4,4,4,5,6,6,7,8,8,9,9</TD></tr><tr><TD >8</TD><TD >0,0,0,1,1,2,2,3,4,4,5,5,8,9</TD></tr><tr><TD >9</TD><TD >0,2,5,8</TD></tr>\n" );
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document.write( "The stem-and-leaf plot is complete. The last thing to do really is to write a key or legend at the bottom of the table to give the reader a way to read the table.<BR>\n" );
document.write( "For instance, you can write 2 | 3 to mean the number 23. The vertical line separates the stem from the leaf. \r<BR>\n" );
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document.write( "===========================================================================\r<BR>\n" );
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document.write( "Part (b)\r<BR>\n" );
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document.write( "As mentioned earlier, we have n = 60 values. \r<BR>\n" );
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document.write( "n/2 = 60/2 = 30<BR>\n" );
document.write( "The median is the midpoint of the values in slot 30 and slot 31.<BR>\n" );
document.write( "Those values are 71 and 72. It might help to break the 6 by 10 block into two 3 by 10 blocks to quickly locate 71 and 72.\r<BR>\n" );
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document.write( "Compute the midpoint: (71+72)/2 = 71.5\r<BR>\n" );
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document.write( "Median = 71.5\r<BR>\n" );
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document.write( "The command named \"MEDIAN\" in spreadsheet software will quickly determine the median. \r<BR>\n" );
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document.write( "Now split the data into two halves<BR>\n" );
document.write( "L = lower half, stuff smaller than median<BR>\n" );
document.write( "U = upper half, stuff larger than median\r<BR>\n" );
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document.write( "L = {<BR>\n" );
document.write( "10,15,17,23,25,32,34,36,41,41<BR>\n" );
document.write( "43,48,52,52,54,55,57,60,60,61<BR>\n" );
document.write( "62,63,64,64,65,67,67,69,70,71<BR>\n" );
document.write( "}\r<BR>\n" );
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document.write( "U = {<BR>\n" );
document.write( "72,74,74,74,75,76,76,77,78,78<BR>\n" );
document.write( "79,79,80,80,80,81,81,82,82,83<BR>\n" );
document.write( "84,84,85,85,88,89,90,92,95,98<BR>\n" );
document.write( "}\r<BR>\n" );
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document.write( "Find the median of set L. That value is 54.5. This is the value of Q1 (aka first quartile).\r<BR>\n" );
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document.write( "The value of Q3 is 80.5 because it is the median of set U.\r<BR>\n" );
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document.write( "Summary:<BR>\n" );
document.write( "lower quartile = Q1 = 54.5<BR>\n" );
document.write( "Median = 71.5<BR>\n" );
document.write( "upper quartile = Q3 = 80.5\r<BR>\n" );
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document.write( "===========================================================================\r<BR>\n" );
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document.write( "Part (c)\r<BR>\n" );
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document.write( "Use the spreadsheet function named COUNTIF to count the number of values smaller than 65.\r<BR>\n" );
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document.write( "The command I typed in was: <font color=blue>=COUNTIF(A1:J6,\"<65\")</font> where A1 is the top left corner of the block of data values. <BR>\n" );
document.write( "The portion A1:J6 means \"look at the rectangle that has A1 in the top left corner and J6 in the bottom right corner\".<BR>\n" );
document.write( "Don't forget about the equal sign up front.\r<BR>\n" );
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document.write( "The result of that command is 24.\r<BR>\n" );
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document.write( "Here is a list of those 24 values that are smaller than 65:<BR>\n" );
document.write( "<TABLE  border = \"1\" cellpadding = \"5\">\n" );
document.write( "<tr><TD >10</TD><TD >15</TD><TD >17</TD><TD >23</TD><TD >25</TD><TD >32</TD></tr><tr><TD >34</TD><TD >36</TD><TD >41</TD><TD >41</TD><TD >43</TD><TD >48</TD></tr><tr><TD >52</TD><TD >52</TD><TD >54</TD><TD >55</TD><TD >57</TD><TD >60</TD></tr><tr><TD >60</TD><TD >61</TD><TD >62</TD><TD >63</TD><TD >64</TD><TD >64</TD></tr>\n" );
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document.write( "This table has 4 rows and 6 columns.\r<BR>\n" );
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document.write( "There are 24 values we want out of 60 values total.\r<BR>\n" );
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document.write( "24/60 = 0.40 = 40% of the scores are less than 65.\r<BR>\n" );
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document.write( "Answer: 40% <BR>\n" );
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