Question 1208080
<font color=black size=3>
Part (i)


Original table
<table border = "1" cellpadding = "5"><tr><td>Marks</td><td>Number of students</td></tr><tr><td>40-44</td><td>7</td></tr><tr><td>45-49</td><td>10</td></tr><tr><td>50-54</td><td>20</td></tr><tr><td>55-59</td><td>f4</td></tr><tr><td>60-64</td><td>f5</td></tr><tr><td>65-69</td><td>6</td></tr><tr><td>70-74</td><td>3</td></tr></table>
Spreadsheet software is strongly recommended.


20% of 75 = 0.20*75 = 15 students have marks between 55 and 59.
This means f4 = 15.


Add up the frequencies in the 2nd column. 
Set this sum equal to 75 so we can determine the value of f5.
7+10+20+f4+f5+6+3 = 75
7+10+20+15+f5+6+3 = 75
61+f5 = 75
f5 = 75-61
f5 = 14



--------------------------------------------------------------------------


Part (ii)


In the previous part we found that 
f4 = 15 and f5 = 14


After replacing f4 and f5 with those values, we now have this grouped frequency table.
<table border = "1" cellpadding = "5"><tr><td>Marks</td><td>Number of students</td></tr><tr><td>40-44</td><td>7</td></tr><tr><td>45-49</td><td>10</td></tr><tr><td>50-54</td><td>20</td></tr><tr><td>55-59</td><td>15</td></tr><tr><td>60-64</td><td>14</td></tr><tr><td>65-69</td><td>6</td></tr><tr><td>70-74</td><td>3</td></tr></table>
Let's introduce a new column which I'll refer to as column m.
m = midpoint of the corresponding class interval
To find the midpoint, add the endpoints and divide by 2.
Example: m = 42 for the first class since (40+44)/2 = 42


Here's what the table looks like now
<table border = "1" cellpadding = "5"><tr><td>Marks</td><td>Number of students</td><td>m</td></tr><tr><td>40-44</td><td>7</td><td>42</td></tr><tr><td>45-49</td><td>10</td><td>47</td></tr><tr><td>50-54</td><td>20</td><td>52</td></tr><tr><td>55-59</td><td>15</td><td>57</td></tr><tr><td>60-64</td><td>14</td><td>62</td></tr><tr><td>65-69</td><td>6</td><td>67</td></tr><tr><td>70-74</td><td>3</td><td>72</td></tr></table>
The midpoint is the best representative mark from each class interval.
Multiply the frequency value (f) with its corresponding midpoint (m).
This will form a new column which I'll label as f*m.
For example, f*m = 7*42 = 294 is the first item in this new column.


<table border = "1" cellpadding = "5"><tr><td>Marks</td><td>Number of students</td><td>M = midpoint</td><td>f*m</td></tr><tr><td>40-44</td><td>7</td><td>42</td><td>294</td></tr><tr><td>45-49</td><td>10</td><td>47</td><td>470</td></tr><tr><td>50-54</td><td>20</td><td>52</td><td>1040</td></tr><tr><td>55-59</td><td>15</td><td>57</td><td>855</td></tr><tr><td>60-64</td><td>14</td><td>62</td><td>868</td></tr><tr><td>65-69</td><td>6</td><td>67</td><td>402</td></tr><tr><td>70-74</td><td>3</td><td>72</td><td>216</td></tr></table>
Add up the values in this new column to get
294+470+1040+855+868+402+216 = 4145


Then divide this over the total number of people (75) to get 4145/75 = 55.266667 which is the approximate mean.
The 6's go on forever but we have to round at some point.



--------------------------------------------------------------------------



Answers:<ol type="i"><li>f4 = 15 and f5 = 14</li><li>mean = 55.266667 approximately</li></ol>
</font>