Question 1206561
<font color=black size=3>
Part (a)


910 total students distributed among 14 classes
910/14 = <font color=red>65</font> students is the average class size.


--------------------------------------------------------------------------
Part (b)


Let's label the four class types as A,B,C,D
A = 30 students
B = 50 students
C = 80 students
D = 100 students


There is one copy of A and C each, 8 copies of B, and 4 copies of D.
P(A) = 1/14
P(B) = 8/14 = 4/7
P(C) = 1/14
P(D) = 4/14 = 2/7


Therefore the PDF is
<table border = "1" cellpadding = "5"><tr><td>x</td><td>P(x)</td></tr><tr><td>30</td><td>1/14</td></tr><tr><td>50</td><td>4/7</td></tr><tr><td>80</td><td>1/14</td></tr><tr><td>100</td><td>2/7</td></tr></table>
PDF = probability density function


--------------------------------------------------------------------------
Part (c)


Now to fill out the x*P(x) column.
Multiply each x item with its paired P(x) probability.
Example: 30*(1/14) = 30/14 = 15/7
<table border = "1" cellpadding = "5"><tr><td>x</td><td>P(x)</td><td>x*P(x)</td></tr><tr><td>30</td><td>1/14</td><td>15/7</td></tr><tr><td>50</td><td>4/7</td><td>200/7</td></tr><tr><td>80</td><td>1/14</td><td>40/7</td></tr><tr><td>100</td><td>2/7</td><td>200/7</td></tr></table>
Adding up the items in the x*P(x) column then gets us:
mu = (15/7)+(200/7)+(40/7)+(200/7)
= (15+200+40+200)/7
= 455/7
= <font color=red>65</font>


Therefore <font color=red>mu = 65</font> is the mean or expected value.
We arrive at the same answer as part (a).


--------------------------------------------------------------------------
Part (d)


We'll use mu = 65 to find the variance.


Introduce a new column called (X-mu)^2*P(X)
The naming of this column should be self-explanatory. If not then please let me know.
<table border = "1" cellpadding = "5"><tr><td>X</td><td>P(X)</td><td>X*P(X)</td><td>(X-mu)^2*P(X)</td></tr><tr><td>30</td><td>1/14</td><td>15/7</td><td>175/2</td></tr><tr><td>50</td><td>4/7</td><td>200/7</td><td>900/7</td></tr><tr><td>80</td><td>1/14</td><td>40/7</td><td>225/14</td></tr><tr><td>100</td><td>2/7</td><td>200/7</td><td>350</td></tr></table>
variance = sum of the (x-mu)^2*P(x) values
variance = (175/2)+(900/7)+(225/14)+350
variance = 4075/7


As the last set of steps:
SD = standard deviation
SD = sqrt(variance)
SD = sqrt(4075/7)
SD = <font color=red>24.13</font> approximately when rounding to 2 decimal places.


The standard deviation is a measure of spread. The larger the standard deviation, the more spread out the distribution will be. 
The same applies to the variance. 
The smallest both items can be is 0, and only occurs when all values in the set are the same. 
</font>